Data Bank
NEA/DB/DOC(2014)1
www.oecd-nea.org
General Description
of Fission Observables
JEFF Report 24
GEF Model
Data Bank NEA/DB/DOC(2014)1
General Description of Fission Observables
GEF Model
Karl-Heinz Schmidt
Beatriz Jurado
CENBG, CNRS/IN2P3, Gradignan, France
Charlotte Amouroux
CEA, DSM-Saclay, France
June 2014
© OECD 2014
NUCLEAR ENERGY AGENCY
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Foreword
Foreword
The Joint Evaluated Fission and Fusion (JEFF) Project is a collaborative effort among
the member countries of the OECD Nuclear Energy Agency (NEA) Data Bank to develop
a reference nuclear data library. The JEFF library contains sets of evaluated nuclear data,
mainly for fission and fusion applications; it contains a number of different data types,
including neutron and proton interaction data, radioactive decay data, fission yield data
and thermal scattering law data.
The General fission (GEF) model is based on novel theoretical concepts and ideas
developed to model low energy nuclear fission. The GEF code calculates fission-fragment
yields and associated quantities (e.g. prompt neutron and gamma) for a large range of
nuclei and excitation energy. This opens up the possibility of a qualitative step forward
to improve further the JEFF fission yields sub-library.
This report describes the GEF model which explains the complex appearance of fission
observables by universal principles of theoretical models and considerations on the basis
of fundamental laws of physics and mathematics. The approach reveals a high degree
of regularity and provides a considerable insight into the physics of the fission process.
Fission observables can be calculated with a precision that comply with the needs for
applications in nuclear technology. The relevance of the approach for examining the
consistency of experimental results and for evaluating nuclear data is demonstrated.
c OECD 2014
General description of fission, GEF model, 3
Acknowledgements
Acknowledgements
Developments for the GEF code have been supported by the European Commission within
the Sixth Framework Programme through EFNUDAT (project No. 036434) and within
the Seventh Framework Programme through Fission-2010-ERINDA (project No. 269499),
and by the OECD Nuclear Energy Agency. We thank Mr R. Capote for drawing our
attention to recent data on prompt-neutron spectra from fission induced by energetic
neutrons. Special thanks go to Mr E. Dupont who incited the work on this report and
followed it with much interest and many helpful remarks.
4
c OECD 2014
General description of fission, GEF model, Table of contents
Table of contents
1 Introduction
1.1 Core of the model . . . . . . . . . . . . . . . . . . . . .
1.2 Additional ingredients . . . . . . . . . . . . . . . . . .
1.3 Understanding of the fission process . . . . . . . . . . .
1.4 Further developments . . . . . . . . . . . . . . . . . . .
1.5 Complementing, estimating and evaluating nuclear data
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2 Concept of the model
2.1 Completeness and generality . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 General features of quantum mechanics . . . . . . . . . . . . . . . . . . .
2.3 General features of stochastic processes . . . . . . . . . . . . . . . . . . .
2.4 General features of statistical mechanics . . . . . . . . . . . . . . . . . .
2.5 Topological properties of the nuclear potential in multi-dimensional space
2.6 Exploiting empirical knowledge . . . . . . . . . . . . . . . . . . . . . . .
2.7 Monte-Carlo method: Correlations and dependencies . . . . . . . . . . .
2.8 Range of validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Special developments
3.1 Systematics of fission barriers . . . . . .
3.2 Nuclear level densities . . . . . . . . . .
3.3 Empirical fragment shells . . . . . . . . .
3.4 Charge polarisation . . . . . . . . . . . .
3.5 Quantum oscillators of normal modes . .
3.6 Fission-fragment angular momentum . .
3.7 Energetics of the fission process . . . . .
3.7.1 From saddle to scission . . . . . .
3.7.2 Fully accelerated fragments . . .
3.8 Even-odd effects . . . . . . . . . . . . . .
3.8.1 Z distribution . . . . . . . . . . .
3.8.2 N distribution . . . . . . . . . . .
3.8.3 Total kinetic energy . . . . . . . .
3.9 Spontaneous fission . . . . . . . . . . . .
3.10 Emission of prompt neutrons and prompt
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gammas
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4 Particle-induced fission
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5 Multi-chance fission
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General description of fission, GEF model, 5
Table of contents
6 Parameter values
6.1 Positions of the fission channels . .
6.2 Widths of the fission channels . . .
6.3 Strength of the fragment shells . . .
6.4 Fragment deformation . . . . . . .
6.5 Charge polarisation . . . . . . . . .
6.6 Energies and temperatures . . . . .
6.6.1 Temperatures . . . . . . . .
6.6.2 Excitation energy at scission
6.6.3 Deformation energy . . . . .
6.6.4 Tunneling . . . . . . . . . .
6.7 Concluding remarks . . . . . . . . .
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7 Uncertainties and covariances
8 Assessment
8.1 Fission probability . . . . . . . . . . . . . . . . . . . . . . .
8.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .
8.1.2 Formulation of the fission probability . . . . . . . . .
8.1.3 Comparison with experimental data . . . . . . . . . .
8.1.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . .
8.1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . .
8.2 Fission-fragment yields . . . . . . . . . . . . . . . . . . . . .
8.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .
8.2.2 Experimental techniques . . . . . . . . . . . . . . . .
8.2.3 Mass distributions . . . . . . . . . . . . . . . . . . .
8.2.4 Deviations . . . . . . . . . . . . . . . . . . . . . . . .
8.2.5 Charge polarisation and emission of prompt neutrons
8.2.6 Nuclide distributions . . . . . . . . . . . . . . . . . .
8.2.7 Energy dependence . . . . . . . . . . . . . . . . . . .
8.2.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . .
8.2.9 Conclusion and outlook . . . . . . . . . . . . . . . . .
8.3 Isomeric yields . . . . . . . . . . . . . . . . . . . . . . . . .
8.4 Prompt-neutron multiplicities . . . . . . . . . . . . . . . . .
8.4.1 System dependence . . . . . . . . . . . . . . . . . . .
8.4.2 Energy dependence . . . . . . . . . . . . . . . . . . .
8.4.3 Fragment-mass dependence . . . . . . . . . . . . . .
8.4.4 Multiplicity distributions . . . . . . . . . . . . . . . .
8.4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . .
8.5 Prompt-neutron energies . . . . . . . . . . . . . . . . . . . .
8.5.1 Key systems . . . . . . . . . . . . . . . . . . . . . . .
8.5.2 Energy dependence . . . . . . . . . . . . . . . . . . .
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General description of fission, GEF model, Table of contents
8.6
8.7
Prompt-gamma emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
Fragment kinetic energies . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
179
9 Data for application
9.1 Decay heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
9.2 Delayed neutrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
10 Validation and evaluation of nuclear data
10.1 Indications for a target contaminant . . . . . . . .
10.2 An inconsistent mass identification . . . . . . . .
10.3 A problem of energy conservation in 252 Cf(sf) . .
10.4 Complex properties of fission channels . . . . . .
10.5 Data completion and evaluation . . . . . . . . . .
10.5.1 Mathematical procedure in two dimensions
10.5.2 Two examples . . . . . . . . . . . . . . . .
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184
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11 Conclusion
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References
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List of Figures
1
2
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Validity range of the GEF model . . . . . . . . . . . . . . . . . . .
Microscopic and macroscopic potential in fission direction . . . . . .
Empirical correction to the fission barrier . . . . . . . . . . . . . . .
Overview on fission barriers, part 1 . . . . . . . . . . . . . . . . . .
Overview on fission barriers, part 2 . . . . . . . . . . . . . . . . . .
Systematics of fission-fragment distributions . . . . . . . . . . . . .
Mean neutron and proton number of heavy mass peak . . . . . . . .
Potential energy for mass-asymmetric distortions . . . . . . . . . .
Nuclide distribution on the N − Z plane . . . . . . . . . . . . . . .
Mean nuclear charge for fixed fragment mass . . . . . . . . . . . . .
Widths of the asymmetric fission channels . . . . . . . . . . . . . .
Energetics of the fission process . . . . . . . . . . . . . . . . . . . .
Variation of the prompt-neutron yield with excitation energy . . . .
A − Ekin distribution . . . . . . . . . . . . . . . . . . . . . . . . . .
TKE distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Potential energy on the fission path . . . . . . . . . . . . . . . . . .
Probabilities of fission chances . . . . . . . . . . . . . . . . . . . . .
Excitation energies at fission . . . . . . . . . . . . . . . . . . . . . .
Covariance matrix of fragment mass yields . . . . . . . . . . . . . .
Uncertainties of mass yields from perturbed-parameter calculations
c OECD 2014
General description of fission, GEF model, .
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Table of contents
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Reduction of fission-decay width . . . . . . . . . . . . . . . . . .
Benchmark of fission probabilities, part 1 . . . . . . . . . . . . .
Benchmark of fission probabilities, part 2 . . . . . . . . . . . . .
Benchmark of fission probabilities, part 3 . . . . . . . . . . . . .
Benchmark of fission probabilities, part 4 . . . . . . . . . . . . .
Benchmark of fission probabilities, part 5 . . . . . . . . . . . . .
Benchmark of fission probabilities, part 6 . . . . . . . . . . . . .
Benchmark of fission probabilities, part 7 . . . . . . . . . . . . .
Benchmark of fission probabilities, part 8 . . . . . . . . . . . . .
Benchmark of fission probabilities, part 9 . . . . . . . . . . . . .
Mass distributions, spontaneous fission, part 1, linear scale . . .
Mass distributions, spontaneous fission, part 1, logarithmic scale
Mass distributions, spontaneous fission, part 2, linear scale . . .
Mass distributions, spontaneous fission, part 2, logarithmic scale
Mass distributions, spontaneous fission, part 3, linear scale . . .
Mass distributions, spontaneous fission, part 3, logarithmic scale
Mass distributions, spontaneous fission, part 4, linear scale . . .
Mass distributions, spontaneous fission, part 4, logarithmic scale
Mass distributions, (nth , f ), part 1, linear scale . . . . . . . . . .
Mass distributions, (nth , f ), part 1, logarithmic scale . . . . . .
Mass distributions, (nth , f ), part 2, linear scale . . . . . . . . . .
Mass distributions, (nth , f ), part 2, logarithmic scale . . . . . .
Mass distributions, (nth , f ), part 3, linear scale . . . . . . . . . .
Mass distributions, (nth , f ), part 3, logarithmic scale . . . . . .
Mass distributions, (nf ast , f ), part 1, linear scale . . . . . . . . .
Mass distributions, (nf ast , f ), part 1, logarithmic scale . . . . . .
Mass distributions, (nf ast , f ), part 2, linear scale . . . . . . . . .
Mass distributions, (nf ast , f ), part 2, logarithmic scale . . . . . .
Mass distributions, (nf ast , f ), part 3, linear scale . . . . . . . . .
Mass distributions, (nf ast , f ), part 3, logarithmic scale . . . . . .
Mass distributions, (nf ast , f ), part 4, linear scale . . . . . . . . .
Mass distributions, (nf ast , f ), part 4, logarithmic scale . . . . . .
Mass distributions, En = 14 MeV, part 1, linear scale . . . . . .
Mass distributions, En = 14 MeV, part 1, logarithmic scale . . .
Mass distributions, En = 14 MeV, part 2, linear scale . . . . . .
Mass distributions, En = 14 MeV, part 2, logarithmic scale . . .
Charge density of post-neutron fragments . . . . . . . . . . . . .
Isobaric Z distributions for 238 U(nth ,f), part 1, logarithmic scale
Isobaric Z distributions for 238 U(nth ,f), part 2, logarithmic scale
Isobaric Z distributions for 238 U(nth ,f), part 3, logarithmic scale
Isobaric Z distributions for 238 U(nth ,f), part 4, logarithmic scale
Isobaric Z distributions for 238 U(nth ,f), part 5, logarithmic scale
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Isobaric Z distributions for 238 U(nth ,f), part 1, linear scale . . . . .
Isobaric Z distributions for 238 U(nth ,f), part 2, linear scale . . . . .
Isobaric Z distributions for 238 U(nth ,f), part 3, linear scale . . . . .
Isobaric Z distributions for 238 U(nth ,f), part 4, linear scale . . . . .
Isobaric Z distributions for 238 U(nth ,f), part 5, linear scale . . . . .
Evolution of Y (A = 115)/Y (A = 140) with excitation energy . . . .
Energy dependence of mass yields in 235 U(n,f) . . . . . . . . . . . .
Energy dependence of mass yields in 239 Pu(n,f) . . . . . . . . . . .
Mass yields for 235 U(n,f), En = 4 MeV and 8 MeV . . . . . . . . . .
Mass yields for 239 Pu(n,f), En = 4 MeV and 8 MeV . . . . . . . . .
Mass yields for 235 U(n,f) and 239 Pu(n,f), En = 14 MeV . . . . . . .
Chi-squared deviations for mass distributions . . . . . . . . . . . . .
Isomeric ratios for 239 Pu(nth ,f). . . . . . . . . . . . . . . . . . . . .
Isomeric ratios for odd-Z compound nuclei . . . . . . . . . . . . . .
Isomeric ratios for even-Z compound nuclei . . . . . . . . . . . . . .
Isomeric ratio for Sb isotopes. . . . . . . . . . . . . . . . . . . . . .
Isomeric ratio for 135 Xe . . . . . . . . . . . . . . . . . . . . . . . . .
Isomeric ratios for light fragments . . . . . . . . . . . . . . . . . . .
Energy dependence of isomeric ratios for 133 Xe and 135 Xe . . . . . .
Isomeric ratio for 134 I from photofission . . . . . . . . . . . . . . . .
Systematics of prompt-neutron multiplicities for spontaneous fission
Systematics of prompt-neutron multiplicities for nth -induced fission
Energy dependence of mean prompt-neutron multiplicities . . . . .
Variation of the mass-dependent prompt neutron yield with E ∗ . . .
Mass dependent prompt-neutron yield in 252 Cf(sf) . . . . . . . . . .
Distribution of prompt-neutron multiplicities . . . . . . . . . . . . .
Prompt-neutron spectra for 235 U(nth ,f) and 252 Cf(sf) . . . . . . . . .
Prompt-neutron spectrum for 239 Pu(nth ,f) . . . . . . . . . . . . . .
Prompt-fission-neutron spectrum for 240 Pu(sf) . . . . . . . . . . . .
Prompt-neutron multiplicities versus neutron direction . . . . . . .
Mean prompt-neutron yield as a function of TKE . . . . . . . . . .
Mean prompt-neutron yield as a function of TKE for 252 Cf(sf) . . .
Prompt-neutron spectrum from 232 Th(n,f), En = 2.9 MeV . . . . .
Prompt-neutron spectrum from 232 Th(n,f), En = 14.7 MeV . . . . .
Prompt-neutron spectrum from 233 U(nth ,f) . . . . . . . . . . . . . .
Prompt-neutron spectrum from 235 U(n,f), En = 100 K . . . . . . .
Prompt-neutron spectrum from 235 U(nth ,f) . . . . . . . . . . . . . .
Prompt-neutron spectrum from 238 U(n,f), En = 2.9 MeV . . . . . .
Prompt-neutron spectrum from 238 U(n,f), En = 5 MeV . . . . . . .
Prompt-neutron spectrum from 238 U(n,f), En = 6 MeV . . . . . . .
Prompt-neutron spectrum from 238 U(n,f), En = 7 MeV . . . . . . .
Prompt-neutron spectrum from 238 U(n,f), En = 10 MeV . . . . . .
c OECD 2014
General description of fission, GEF model, .
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9
Table of contents
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10
Prompt-neutron spectrum from 238 U(n,f), En = 13.2 MeV . . . . .
Prompt-neutron spectrum from 238 U(n,f), En = 14.7 MeV . . . . .
Prompt-neutron spectrum from 239 Pu(nth ,f) . . . . . . . . . . . . .
Prompt-neutron spectrum from 240 Pu(sf) . . . . . . . . . . . . . . .
Prompt-neutron spectrum from 242 Pu(sf) . . . . . . . . . . . . . . .
Prompt-neutron spectrum from 241 Am(n,f), En = 2.9 MeV . . . . .
Prompt-neutron spectrum from 241 Am(n,f), En = 4.5 MeV . . . . .
Prompt-neutron spectrum from 241 Am(n,f), En = 14.6 MeV . . . .
Prompt-neutron spectrum from 242 Am(nth ,f) . . . . . . . . . . . . .
Prompt-neutron spectrum from 243 Am(n,f), En = 2.9 MeV . . . . .
Prompt-neutron spectrum from 243 Am(n,f), En = 4.5 MeV . . . . .
Prompt-neutron spectrum from 243 Am(n,f), En = 14.6 MeV . . . .
Prompt-neutron spectrum from 243 Cm(nth ,f) . . . . . . . . . . . . .
Prompt-neutron spectrum from 244 Cm(sf) . . . . . . . . . . . . . .
Prompt-neutron spectrum from 245 Cm(nth ,f) . . . . . . . . . . . . .
Prompt-neutron spectrum from 246 Cm(sf) . . . . . . . . . . . . . .
Prompt-neutron spectrum from 248 Cm(sf) . . . . . . . . . . . . . .
Prompt-gamma spectrum for 235 U(nth ,f) . . . . . . . . . . . . . . .
Prompt-gamma spectrum for 252 Cf(sf) . . . . . . . . . . . . . . . .
Mass-dependent mean kinetic energy for 233 U(nth ,f) . . . . . . . . .
Mass-dependent mean kinetic energy for 235 U(nth ,f) . . . . . . . . .
Mass-dependent mean kinetic energy for 239 Pu(nth ,f) . . . . . . . .
Mass-dependent TKE before prompt-neutron emission for 252 Cf(sf)
Mass-dependent pre-neutron TKE for 232 Th(n,f) . . . . . . . . . . .
Pre-neutron TKE distribution for 240 Pu(nth ,f) and 240 Pu(sf) . . . .
Variance of the TKE distribution for neutron-induced fission . . . .
Mass-dependent width of the TKE distribution for 252 Cf(sf) . . . .
Width of the TKE distribution for 233 U(nth ,f) . . . . . . . . . . . .
TKE as a function of the incident-neutron energy for 235 U(n,f) . . .
Total decay heat for 235 U(nth ,f) . . . . . . . . . . . . . . . . . . . .
Influence of the odd-even effect on the delayed-neutron yield . . . .
Delayed-neutron yield for 235 U(n,f), En = 1 MeV . . . . . . . . . .
Delayed-neutron yields for 237 Np(n,f), 235 U(n,f) and 238 U(n,f) . . . .
Calculated mass yields for 235 U(n,f), En = thermal and 5 MeV . .
Delayed neutron yield up to En = 14 MeV . . . . . . . . . . . . . .
Evidence for a 239 Pu contaminant in a 237 Np target . . . . . . . . .
Inconsistency of mass and TKE for 232 Th(n,f), En = 2.9 MeV . . .
Matching measured 235 U(nth ,f) mass yields with GEF results . . . .
Matching measured 241 Pu(nf ast ,f) mass yields with GEF results . .
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General description of fission, GEF model, Table of contents
List of Tables
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
RMS deviation between different sets of fission barriers . . . . . . .
Fission barriers used in GEF, part 1 . . . . . . . . . . . . . . . . . .
Fission barriers used in GEF, part 2 . . . . . . . . . . . . . . . . . .
Stiffness coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . .
Strengths of the fragments shells near the outer fission barrier. . . .
Effective temperature parameter for tunneling . . . . . . . . . . . .
Standard deviations of perturbed parameter values . . . . . . . . .
Measured and evaluated mass distributions, part 1 . . . . . . . . . .
Measured and evaluated mass distributions, part 2 . . . . . . . . . .
Measured and evaluated mass distributions, part 3 . . . . . . . . . .
First-chance probability for 235 U(n,f), En = 8 MeV and 14 MeV. . .
First-chance probability for 239 Pu(n,f), En = 8 MeV and 14 MeV. .
Mean TKE before prompt-neutron emission for well known systems
Weights of fission modes for 239 Pu(nth ,f) and 240 Pu(sf) . . . . . . .
Delayed-neutron yields . . . . . . . . . . . . . . . . . . . . . . . . .
Prompt-neutron multiplicity of the system 252 Cf(sf) . . . . . . . . .
Yields of fission channels for the system 252 Cf(sf) . . . . . . . . . .
c OECD 2014
General description of fission, GEF model, .
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11
Introduction
1
1.1
Introduction
Core of the model
The phenomena related to nuclear fission result from many different processes. For many
of these, there exist very elaborate models, for example for the capture of an incoming
particle in a target nucleus and for the emission of neutrons and gamma radiation from
an excited nucleus. However, the modelling of the re-ordering of the nucleons from an
excited mono-nucleus into two (or eventually more) fragments is still a challenge for
nuclear theory. Estimating the properties of the fission fragments with the high quality
required for applications in nuclear technology still relies on empirical models [1]. The
composition of the fragments in A and Z determines the starting points of the radioactive
decay chains and defines the decay-heat production, their excitation energies determine
the characteristics of prompt-neutron and prompt-gamma emission. The major problem is
that a fissioning nucleus is an open system that evolves from a quasi-bound configuration
to a continuum of possible configurations on the fission path, finally forming hundreds
of different fragments with continuous distributions of different shapes, kinetic energies,
excitation energies and angular momenta. One of the most advanced approaches for
modelling low-energy nuclear fission describes the fission process by a numerical solution
of the Langevin equations [2, 3] with eventually further approximations e.g. neglecting
the influence of inertia on the dynamics [4]. A subspace of collective variables that is
restricted by the limited available computing power is explicitly considered, while the
coupling to most internal degrees of freedom is replaced by a heat bath. It is a draw-back
that quantum-mechanical features are not properly considered in this classical approach.
Another one follows the evolution of the fissioning system with quantum-mechanical tools
[5]. However, the inclusion of dissipative processes and phenomena of statistical mechanics
within quantum-mechanical algorithms is still not sufficiently developed [6, 7]. Also these
calculations require very large computer resources.
The general description of fission observables (GEF model) presented in this work
makes use of many theoretical ideas of mostly rather general character, avoiding microscopic calculations with their inherent approximations, e.g. the parameterisation of the
nuclear force, and limitations, e.g. by the high computational needs. The large body of
empirical information is used for developing a general description of the fission process,
which is in good agreement with the empirical data. The theoretical frame assures that
this model is able to provide quantitative predictions of the manifold fission observables
for a wide range of fissioning systems.
The empirical input to the model requires adjusting a number of parameters by performing a large number of calculations for many systems. For this purpose, it is important
that the calculation is relatively fast, allowing for applying fit procedures in order to find
the optimum parameter values.
12
c OECD 2014
General description of fission, GEF model, Introduction
1.2
Additional ingredients
For the calculation of fission observables it is not enough to master the dynamics of the
fission process, starting from an excited compound nucleus and ending with the formation
of two separated nuclei at scission. Also the initial reaction that induced the fission
process, for example the capture of a neutron, must be described, eventually including precompound processes. Furthermore, the competition of particle emission, gamma emission
and fission must be considered, because it determines the relative contributions of different
fissioning systems with different excitation-energy distributions, if the initial excitation
energy is high enough for multi-chance fission to occur. After scission, the fragments may
be highly excited, and, thus, they emit a number of particles, mostly neutrons and gamma
radiation. Also, these processes should be considered in a complete model.
The GEF code aims to provide a complete description including the entrance channel
and the de-excitation of the fragments. This is particularly important for the determination of the optimum parameters of the model, because all available observables should
be included in the fit procedure. For this purpose, the algorithms should be very efficient
in order to assure a short computing time. Therefore, whenever possible, approximations
and analytical descriptions were preferred if they are precise enough not to alter the results beyond the inherent uncertainties of the model. More elaborate models that have
been developed in many cases may easily be implemented in the code, once it will be fully
developed.
1.3
Understanding of the fission process
The general character of the model makes it necessary to establish the systematics of the
variation of the fission observables for different systems and as a function of excitation
energy and to interpret the origin of these features. It will be shown that the basic ideas
of the model are astonishingly powerful. Therefore, the links between the observations
and the ingredients of the model enable extracting valuable information on the physics
of the nuclear-fission process, much better than the inspection of isolated systems or
the direct study of the measurements. An essential advantage provided by the model
is the consistent description of essentially all degrees of freedom, due to the efficient
computational technique of the code.
1.4
Further developments
The GEF model is unique in the sense that it treats the complete fission process with
explicit consideration of a large number of degrees of freedom in a coherent way on physics
ground. Thus it provides the links between the different processes and mechanisms and the
fission observables. Even more, it preserves the correlations between the different degrees
of freedom. It might be very useful for a better understanding of the fission process to
carefully study any discrepancies between the model results and available experimental
data. Moreover, the systematic trends and global features revealed or predicted by the
c OECD 2014
General description of fission, GEF model, 13
Introduction
model may stimulate dedicated experiments and calculations on specific problems with
microscopic models in order to better understand certain aspects of the fission process.
In the application for nuclear data, the GEF code may replace purely parametric
descriptions for many quantities and serve as a realistic, consistent and complete event
generator for transport calculations with dedicated codes like MCNP [8] or FLUKA [9].
1.5
Complementing, estimating and evaluating nuclear data
Since the empirical data are an important input of the model, it is suggestive that it can be
useful for the evaluation of nuclear data. First, the predictions of the model may directly
be used for estimating some fission observables, if no experimental data are available.
The code can also be useful in order to check whether certain experimental results are in
line with or in contradiction with observed trends and systematics. This may lead to an
enhanced or diminished confidence level of these data or eventually stimulate dedicated
experiments for verification. A very useful application may be the exploitation of the code
results for complementing missing data. For this purpose, a special algorithm has been
developed that “fine-tunes” the calculated values in a way that they fit to the available
experimental data. This algorithm is implemented in the computer code MATCH [10].
14
c OECD 2014
General description of fission, GEF model, Concept of the model
2
Concept of the model
2.1
Completeness and generality
During almost seven decades of research, a rather clear, at least qualitative, comprehension
of the different stages of the nuclear-fission process has emerged. These ideas guided us
to establish the following concept of the general fission model, which is the subject of the
present report.
At first, the nucleus needs to leave the first minimum at its ground-state shape, by
passing the fission barrier, which in the actinides consists of two or may be even three
consecutive barriers with a minimum in between. Since tunneling proceeds with a very
low probability, as can be deduced from the long spontaneous-fission half lives, an excited
nucleus has enough time to re-arrange its available energy. The probability for the passage
of the fission barrier increases considerably, if the nucleus concentrates enough of its energy
on the relevant shape degrees of freedom for avoiding tunneling as much as the available
energy allows. The remaining energy, however, can be randomly distributed between the
different states above the barrier without any further restriction, such that the barrier is
passed with maximum possible entropy on the average [11]. For this reason, the fissioning
system has no memory on the configurations before the barrier, except the quantities
that are preserved due to general conservation laws: total energy, angular momentum
and parity. Thus, the starting point of the model is the configuration above the outer
fission barrier.
Beyond the outer barrier, one can define an optimum fission path, consisting of a
sequence of configurations in deformation space with minimum potential energy for a
certain elongation. Although the quantitative determination of this path depends on the
shape parameterisation, this picture is helpful for revealing that the fissioning system
is unbound only with respect to one degree of freedom, the motion in direction of the
fission path. The system is bound with respect to motion in any other direction in
deformation space. The distribution of the collective coordinate is given by the occupation
probability of the states in the respective potential pockets, which eventually may have
more than one minimum. Some of these degrees of freedom which are confined by a
restoring force towards the potential minimum are directly linked to fission observables,
e.g. the mass asymmetry A1 /(A1 + A2 ) or the charge polarisation < Z1 > −ZU CD 1 with
ZU CD = A1 × ZCN /ACN . ACN , ZCN , Ai and Zi are mass and atomic number of the
fissioning system and of one fragment, respectively. The fission-fragment distribution in
Z and A is given by the evolution of the respective collective variables, until the system
reaches the scission configuration. Under the influence of dissipation and inertial forces,
the value of the respective collective variable is the integral result of the forces acting on
the whole fission path.
1
For a continuous tracking of these degrees of freedom, suitable prescriptions must be defined that generalise these values that are defined for the separated fragments to the respective deformation parameters
of the system on the fission path before scission.
c OECD 2014
General description of fission, GEF model, 15
Concept of the model
From a theoretical model, one requires that the evolution of the fissioning system
is fully described, considering all degrees of freedom, their dependencies and their correlations. Current microscopic models, either classical or quantum-mechanical ones, do
not meet this requirement, because they only consider a restricted number of degrees of
freedom. Statistical models applied at the saddle or at the scission point are not suited
neither, because they neglect the dynamical aspects of the fission process.
The general fission model is a compromise that does not eliminate any of the complex
features of the fission process by far-reaching approximations or restrictions from the
beginning on. Instead, it makes use of a number of generally valid physics laws and
characteristics that allow reducing the computing expenses to an affordable level.
2.2
General features of quantum mechanics
The observables from low-energy fission show strong manifestations of quantum-mechanical
effects like the contributions of the different fission modes to the fission-fragment mass
and total-kinetic-energy distributions that are related to nuclear shell effects and the considerable enhancement of even-Z fission fragments that are related to pairing correlations.
These quantum-mechanical features are responsible for great part of the complexity of
nuclear fission, and, thus, they considerably complicate the theoretical description of the
fission process. The GEF model exploits a long-known general property of quantum mechanical wave functions in a strongly deformed potential in order to simplify this problem
considerably.
When the two-centre shell model became available, it was possible to study the singleparticle structure in a di-nuclear potential with a necked-in shape. Investigations of
Mosel and Schmitt [12] revealed that the single-particle structure in the vicinity of the
outer fission barrier already resembles very much the coherent superposition of the singleparticle levels in the two separated fragments after fission. They explained this result
by the general quantum-mechanical feature that wave functions in a slightly necked-in
potential are already essentially localized in the two parts of the system. This feature is a
direct consequence of the necking, independent from the specific shape parameterisation.
This finding immediately leads to the expectation that the shells on the fission path
that are responsible for the complex structure of fission modes are essentially given by
the fragment shells. Potential-energy surfaces of fissioning systems calculated with the
macroscopic-microscopic approach (e.g. ref. [13]) support this assumption.
As a consequence, the shell effects on the fission path can be approximately considered
as the sum of the shell effects in the proton- and neutron-subsystems of the light and the
heavy fission fragment. Thus, these shells do not primarily depend on the fissioning
system but on the number of neutrons and protons in the two fission fragments. However,
these shells may be substantially different from the shell effects of the fragments in their
ground state, because the nascent fragments in the fissioning dinuclear system might be
strongly deformed due to the interaction with the complementary fragment.
Other quantum-mechanical features to be considered are tunneling in spontaneous
16
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fission and the uncertainty principle that induces fluctuations in the nuclide distributions
of the fission fragments and that is assumed to generate their angular momenta.
2.3
General features of stochastic processes
Stochastic calculations revealed that, depending on the nature of the collective degree of
freedom considered, dynamical effects induce a kind of memory on the fission trajectory
due to the influence of dissipation and inertial forces [14]. The corresponding characteristic
memory time determines, after which time a specific coordinate value is forgotten and
how long it takes for this coordinate to adjust to modified conditions. This means that
the distribution of a specific observable is essentially determined by the properties of
the system, for example the potential-energy surface, at an earlier stage. According
to experimental observations [15] and theoretical studies [16], the mass distribution is
essentially determined already way before reaching scission. The memory of the chargepolarisation degree of freedom has been found to be much shorter [17, 18].
As a practical consequence, it is assumed in the GEF model that the measured distribution of a specific fission observable essentially maps a kind of effective potential, the
system was exposed to by the characteristic memory time before reaching scission. In
other words, the effective potential that is extracted from the measured distribution of a
fission observable, implicitly includes the influence of dynamical effects.
2.4
General features of statistical mechanics
The transformation of energy between potential energy, intrinsic and collective excitations as well as kinetic energy is another very important aspect of the nuclear-fission
process. It determines the partition of the fission Q value (plus eventually the initial excitation energy of the fissioning system) between kinetic and excitation energy of the final
fragments. Moreover, the division of the total excitation energy between the fragments
is of considerable interest, because it induces a shift of the isotopic distributions from
the primary fragments by neutron evaporation towards less neutron-rich isotopes. It has
also been noticed that the shape of the mass-dependent prompt-neutron yields strongly
affects the mass values of the fission products deduced from kinematical double-energy
measurements [19].
The GEF model takes advantage of the general laws of statistical mechanics, which
govern the energetics of any object, independently of its size. In particular, statistical
mechanics requires that the available energy tends to be distributed among the accessible
degrees of freedom in equal share during the dynamical evolution of the system. This
general law provides an invaluable estimation of the evolution of the intrinsic excitation
energies and the population of the available states in the nascent fragments during the
fission process with little computational expense.
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General description of fission, GEF model, 17
Concept of the model
2.5
Topological properties of the nuclear potential in multidimensional space
The shape-dependent nuclear potential of the fissioning system possesses a few regions
which have special importance for the fission process. These are the nuclear ground state
and the saddle points at the inner and the outer fission barrier. The topological property
of the ground state is solely defined by the condition that the ground state is the nuclear
state with the largest binding energy. No other condition is imposed on the surrounding
of this state. It is trivial that increasing the binding energy in this state directly translates
in lowering the mass of the nucleus in its ground state. This is different for a saddle point,
which is the lowest threshold that must be passed when moving from the region around
the ground state to the region around the second minimum or from the region around the
second minimum towards scission. In this case, the height of the saddle has to be searched
on a whole path in comparison to all other possible paths across the barrier. In addition,
a local modification of the potential by a bump or a dip, for example by shell effects, does
not have a big effect on the height of the saddle, because the fissioning nucleus will go
around the bump, and it cannot profit from the depth of the dip, because the potential
at its border has changed only little. Myers and Swiatecki expressed these ideas by the
topographical theorem [20]. The topographical theorem is exploited in the GEF model
for deriving a semi-empirical systematics of fission barriers.
2.6
Exploiting empirical knowledge
Many of the ideas outlined in the previous sections establish a link between measured
observables and specific properties of the fissioning system. It is an essential part of the
GEF concept that this empirical information is exploited to assure that the quantitative
results of the model are in best possible agreement with the available experimental data.
For this purpose, the ingredients of the model and its parameter values are adjusted in a
global fit procedure that minimises the deviations from a large set of experimental data of
different kind. Note that the GEF model, in contrast to current empirical models, is not a
direct parameterisation of the observables. Instead, as was mentioned above, it describes
the physics of the fission process, making use of several approximations based on general
physics laws. The quality with which the GEF code is able to reproduce a large body of
data with a moderate number of parameters will give an indication about the validity of
these approximations.
2.7
Monte-Carlo method: Correlations and dependencies
Nuclear fission provides a large number of observables. The correlations between different
kind of observables represent a valuable information on the fission process. Therefore, the
GEF code is designed as a Monte-Carlo code that follows all quantities of the fissioning
systems with their correlations and dependencies. Finally, all observables can be listed
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General description of fission, GEF model, Concept of the model
event by event, which allows the investigation of all kind of correlations. Moreover,
complex experimental filters can easily be implemented.
2.8
Range of validity
According to the concept of the GEF model, the range of validity is not strictly defined.
Technically, the code runs for any heavy nucleus. However, the results of the model are
more reliable for nuclei which are not too far from the region where experimental data
exist. It is recommended not to use the code outside the range depicted in Figure 1 on
the chart of the nuclides.
Figure 1: Validity range of the GEF model
Note: The validity range of the GEF model is marked in yellow. For a detailed description of
the figure see Figure 6.
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General description of fission, GEF model, 19
Special developments
3
3.1
Special developments
Systematics of fission barriers
One of the most critical input parameters of the GEF model is the height of the fission
barrier. That is the energy a nucleus has to invest in order to proceed to fission without
tunneling. Since experimental fission barriers are available for a rather restricted number
of nuclei, only, a model description is needed in order to meet the requirement of the GEF
model for generality.
An elaborate analysis [21] of available experimental data revealed that different theoretical models differ appreciably in their predictions for the average trend of the fissionbarrier height along isotopic chains. Also self-consistent models deviate drastically from
each other.
During the last years, the efforts for developing improved models for the calculation
of fission barriers were intensified, using the macroscopic-microscopic approach [22, 23,
13, 24, 25, 26, 27], the density-functional theory [28, 29] and varieties of Hartree-Fock
methods [30-33]. Still, the results from the different models, in particular in regions,
where no experimental data exist, differ appreciably. Since the fission-barrier height is
the difference of the mass at the saddle-point which defines the fission barrier and the
ground-state mass, it is obvious that the fission-barrier values from these models cannot
be more precise than the values of the ground-state masses, which show typical root-mean
square deviations of at least several 100 keV from the experimental values.
An alternative approach that avoids this problem was used in ref. [34], by estimating
the fission barrier as the sum of the macroscopic fission barrier and the ground-state
shell correction, making use of the topographical theorem [20, 35]. In the GEF model,
we follow this idea, however, we explicitly consider the pairing condensation energies in
the ground state and at the barrier, because they are systematically different. For the
macroscopic part of the fission barrier, we chose the Thomas-Fermi barriers of Myers and
Swiatecki [36] and combined them with the Thomas-Fermi masses of the same authors
[20] for determining the contributions of shells and pairing to the ground-state binding
energy, because these models were found [21] to follow best the isotopic trends of the
experimental masses and fission barriers.
In detail, the macroscopic fission-barrier height for the nucleus with mass number A
and atomic number Z is calculated with the following relations, adapted from ref. [36]:
N =A−Z
(1)
I = (N − Z)/A
(2)
κ = 1.9 + (Z − 80)/75
(3)
S = A2/3 (1 − κI 2 )
X=
20
Z2
A(1 − κI 2 )
(4)
(5)
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General description of fission, GEF model, Special developments
For 30 ≤ X < X1 :
For X ≥ X1 and X ≤ X0 :
F = 0.595553 − 0.124136(X − X1 )
(6)
F = 0.000199749(X0 − X)3
(7)
with X0 = 48.5428 and X1 = 34.15. Finally, the Thomas-Fermi macroscopic fission
barrier is given by:
BfT F = F · S.
(8)
The higher one of the inner (EA ) and the outer (EB ) fission barrier (Bf = max(EA , EB ))
is given by the sum of the macroscopic fission barrier and the negative value of the mimic
plus the microscopic contribution
croscopic contribution to the ground-state mass δEgs
mic
δEf to the binding energy at the respective barrier.
mic
+ δEfmic .
Bf = BfT F − δEgs
(9)
The microscopic contribution to the ground-state mass is the difference of the actual
ground-state mass and the macroscopic mass obtained with the Thomas-Fermi approach
mic
= mgs − mT F . It represents the structural variation of the ground-state mass
δEgs
due to shell correction δUgs and pairing condensation energy δPgs . The topographical
theorem claims that the shell effect at the barrier can be neglected. Therefore, only
the contribution from pairing must be considered: δEfmic = δPf . Best agreement with
the data is obtained by including
the even-odd staggering of the binding energy at the
√
barrier with δPf = −n · 14/ A, n = 0, 1, 2 for odd-odd, odd-mass and even-even nuclei,
respectively.
The available measured fission barriers were used to deduce the following empirical
function, which describes the difference between the inner and the outer barrier height:
EA − EB = 5.40101 − 0.00666175 · Z 3 /A + 1.52531 · 10−6 · (Z 3 /A)2 .
(10)
The result of this procedure was not yet fully satisfactory, because the barriers around
thorium were somewhat overestimated. This discrepancy decreases for lighter and for
heavier elements. Figure 2 illustrates a possible reason for this deviation in a schematic
way: The lower part shows the macroscopic potential, essentially given by the asymmetrydependent surface energy and Coulomb interaction potential, for a lighter (dashed line),
for a medium-heavy (full line) and for a heavier (dot-dashed line) nucleus. The upper
part shows the schematic variation of the shell correction as a function of deformation,
which is assumed to be the same for all nuclei in the region of the heavy nuclei concerned.
The full potential can be assumed as the sum of the macroscopic and the microscopic
potential. The first minimum of the nuclear ground state is deformed in the actinides
considered. The full line in the lower part corresponds to the situation around thorium:
The inner and the outer barriers have about the same height. This situation is realised
when the second minimum is localised near the maximum of the macroscopic potential.
In this situation, the inner and the outer barrier are localised at a deformation, where the
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General description of fission, GEF model, 21
Special developments
Figure 2: Microscopic and macroscopic potential in fission direction
Note: Schematic drawing of the microscopic and the macroscopic potential in fission direction.
See text for details.
macroscopic potential is far from its maximum value. For lighter nuclei, the maximum of
the macroscopic potential moves to larger deformations, closer to the outer barrier, which
becomes the higher one. For heavier nuclei, the maximum of the macroscopic potential
moves to smaller deformations, closer to the inner barrier, which becomes the higher one.
This consideration makes it understandable that the barriers of nuclei around thorium
deviate systematically from the smooth trend expected from the topographical theorem:
They are systematically smaller. This deviation was parametrised by the following correction term:
For 86.5 < Z < 90 :
∆Bf = −0.15(Z − 86.5)
(11)
For 90 ≤ Z < 93 :
∆Bf = −0.15(Z − 86.5) + 0.35(Z − 90)
(12)
∆Bf = −0.15(Z − 86.5) + 0.35(Z − 90) + 0.15(Z − 93)
(13)
∆Bf = −0.15(Z − 86.5) + 0.35(Z − 90) + 0.15(Z − 93) − 0.25(Z − 95)
(14)
For 93 ≤ Z < 95 :
For Z ≥ 95 :
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Figure 3: Empirical correction to the fission barrier
Note: Empirical correction applied to the fission barrier height obtained with the topographical
theorem as a function of the atomic number of the fissioning nucleus.
The resulting function is depicted in Figure 3. In addition to the dip around Z = 90,
which can be considered as a refinement of the topographical theorem, the barrier heights
had to be further increased for the heavier elements in order to better reproduce the measured values. This latter effect, which is the only violation of the topographical theorem
in our description, may be caused by shell effects at the barrier or by a shortcoming of
the Thomas-Fermi barriers for the heaviest elements.
The fission barriers from the description used in the GEF code are compared in Figures
4 and 5 with different empirical and theoretical values. The theoretical values of Goriely
et al. [33] are quite close to the empirical data, whereas the values of P. Möller et al.
[13] 2 deviate strongly in their absolute values and in the isotopic trends. Obviously, the
description used in the GEF code agrees rather well with the empirical data. In particular,
this description can be extrapolated far away from the beta-stable region without any new
assumptions. In several cases, the symbols are even hardly visible because they are covered
by the experimental points.
The even-odd staggering of the fission barrier
√ height is well reproduced by the model
assuming
a pairing-gap parameter ∆ = 14/ A, compared to an averge value of ∆ =
√
12/ A in the nuclear ground state. This may eventually be an evidence for the deformation dependence of the pairing strength [37]. But a stronger pairing at the barrier is also
expected by the systematically higher single-particle level density at the barrier compared
to the ground state due to topological reasons: while the barrier is practically not lowered
by shell effects compared to the macroscopic barrier, the nuclear ground state is almost
generally more bound than the macroscopic ground state, because it is the state with the
2
It is the benefit of ref. [13], compared to most theoretical work, that it presents extensive tables of
calculated barrier parameters of the actinides, which makes the quantitative comparison with this theory
feasible.
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General description of fission, GEF model, 23
Special developments
absolute lowest energy in deformation space. Generally, more binding by shell effects is
related to a lower single-particle level density at the Fermi level. This kind of even-odd
staggering is also present in the theoretical values [13] and [33], although the amplitude
is not large enough.
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Figure 4: Overview on fission barriers, part 1
Note: Height of the inner and the outer fission barrier above the nuclear ground state for isotopes
of protactinium, uranium and neptunium. The description used in the GEF model is compared
with the experimental barriers determined in ref. [38] (marked by ”exp”), with the empirical
barriers given in RIPL3 [39], with the self-consistent theoretical barriers [33] given in RIPL3
(marked by ”Goriely”), and with the macroscopic-microscopic barriers of P. Möller et al. [13].
In addition, the macroscopic fission barriers (BFM S ) from ref. [36] are shown.
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General description of fission, GEF model, 25
Special developments
Figure 5: Overview on fission barriers, part 2
Note: Height of the inner and the outer fission barrier above the nuclear ground state for isotopes
of plutonium, americium and curium. The description used in the GEF model is compared with
the experimental barriers (marked by ”exp”) determined in ref. [38] (full symbols) and other
papers cited in ref. [40] (open symbols), with the empirical barriers given in RIPL3 [39], with
the self-consistent theoretical barriers [33] given in RIPL3 (marked by ”Goriely”), and with the
macroscopic-microscopic barriers of P. Möller et al. [13]. In addition, the macroscopic fission
barriers (BFM S ) from ref. [36] are shown.
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The rms deviations between the different sets of fission barriers shown in Figures 4 and
5 are listed in Table 1. There is a remarkably large deviation between the experimental
data from Bjornholm and Lynn [38] and the recommended values of RIPL 3 [39]. The
best agreement exists between the GEF parameterisation and the experimental values
determined by Bjornholm and Lynn [38]. The rms deviation of 0.2 MeV does not exceed
the estimated uncertainties of the experimental values [38] and is appreciably smaller than
the rms deviations with which the best atomic mass models reproduce the experimental
values.3 Also, the self-consistent barriers of Goriely et al. [33] agree better with the
experimental values of Bjornholm and Lynn than with the recommended values of RIPL
3 [39]. From the figures, one can deduce that the data from ref. [38] and the GEF
parameterisation, which is deduced from the topographical theorem, agree best in the
isotopic trend, while the theoretical values of ref. [33] and the RIPL 3 values show
increasingly discrepant local deviations. The theoretical values of Möller et al. deviate
most strongly from any other set. Considering that the only adjustment of the proposed
description is the application of the simple and well justified global Z-dependent function
shown in Figure 3, experiment and calculation are fully independent in their structural
features and in their global dependency on neutron excess. The good agreement thus
evidences that the barriers obtained with the present approach represent the experimental
values better than the two theoretical models considered or the RIPL-3 recommended
values.
Finally, the fission-barrier values of several heavy nuclei that are used in the GEF code
are listed in Tables 2 and 3.
Table 1: RMS deviation between different sets of fission barriers
exp
RIPL 3
GEF
Goriely
Möller
exp
—
0.43
0.20
0.37
1.1
RIPL 3
0.43
—
0.46
0.46
1.0
GEF Goriely
0.20
0.37
0.46
0.46
—
0.38
0.38
—
1.1
1.0
Möller
1.1
1.0
1.1
1.0
—
Note: The table lists the RMS deviations in MeV between the different sets of fission barriers
shown in Figures 4 and 5. References are given in the Figure captions. The typical uncertainty
of the experimental values is 0.2 to 0.3 MeV.
3
According to ref. [41], the most precise and robust nuclear mass predictions are given by the DufloZuker model [42], which gives an rms deviation of 373 keV.
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Special developments
Table 2: Fission barriers used in GEF, part 1
N
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
Th
5.75/5.87
5.84/6.01
5.64/5.85
5.70/5.96
5.67/5.98
5.78/6.13
5.70/6.09
5.91/6.34
5.72/6.19
5.84/6.35
5.65/6.19
5.87/6.46
5.56/6.18
5.71/6.37
5.19/5.89
Pa
4.99/4.79
5.53/5.39
5.25/5.15
5.64/5.60
5.44/5.45
5.88/5.93
5.91/6.43
6.03/6.18
5.60/5.70
6.03/6.18
5.72/5.91
6.06/6.30
5.48/5.84
5.77/6.17
5.48/5.84
5.77/6.17
5.28/5.72
U
4.12/3.57
4.45/3.95
4.54/4.10
4.84/4.46
5.00/4.67
5.35/5.07
5.37/5.15
5.85/5.67
5.69/5.56
5.95/5.87
5.76/5.73
6.04/6.06
5.64/5.71
5.82/5.94
5.44/5.59
5.63/5.83
5.19/5.43
5.29/5.57
5.63/5.95
Np
3.41/2.46
4.18/3.29
4.07/3.25
4.67/3.91
4.64/3.94
5.22/4.58
5.24/4.66
5.81/5.29
5.60/5.13
6.08/5.66
5.70/5.34
6.19/5.88
5.68/5.43
5.97/5.76
5.63/5.47
5.98/5.87
5.36/5.30
5.77/5.75
5.04/5.07
5.96/6.03
Pu
3.20/1.83
3.48/2.18
3.81/2.57
4.30/3.13
4.37/3.27
4.79/3.75
4.89/3.92
5.43/4.52
5.37/4.53
5.80/5.01
5.65/4.92
6.08/5.41
5.70/5.09
5.97/5.41
5.61/5.10
5.89/5.44
5.49/5.09
5.65/5.31
5.30/5.01
5.34/5.10
5.11/4.91
Note: Height of first and second barrier used in the GEF code in MeV.
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Table 3: Fission barriers used in GEF, part 2
N
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
Am
4.35/2.73
4.28/2.74
4.98/3.51
5.02/3.62
5.75/4.42
5.67/4.41
6.23/5.03
5.97/4.84
6.50/5.43
6.05/5.04
6.39/5.45
5.90/5.01
6.30/5.48
5.73/4.96
5.99/5.28
5.50/4.84
5.66/5.06
5.01/4.47
5.34/4.84
Cm
3.77/1.67
4.05/2.03
4.45/2.51
4.60/2.74
5.16/3.37
5.21/3.49
5.70/4.06
5.64/4.07
6.10/4.60
5.77/4.34
6.08/4.71
5.82/4.52
6.19/4.95
5.82/4.65
6.07/4.96
5.69/4.64
5.71/4.72
5.18/4.25
5.11/4.24
4.64/3.83
Bk
3.64/1.11
4.27/1.82
4.41/2.04
5.11/2.82
5.12/2.92
5.69/3.57
5.63/3.58
6.16/4.19
5.77/3.88
6.23/4.41
5.95/4.19
6.51/4.83
6.05/4.44
6.41/4.87
5.94/4.46
6.02/4.61
5.26/3.91
5.35/4.07
4.70/3.48
4.85/3.69
4.51/3.41
Cf
3.89/0.89
4.12/1.21
4.68/1.85
4.77/2.04
5.22/2.58
5.27/2.70
5.70/3.22
5.57/3.17
5.94/3.62
5.85/3.61
6.37/4.21
6.09/4.00
6.36/4.35
6.02/4.08
6.05/4.18
5.48/3.68
5.43/3.70
4.95/3.29
4.91/3.31
4.46/2.93
4.03/3.17
4.41/3.01
Es
3.95/0.46
4.61/1.21
4.59/1.28
5.22/2.01
5.18/2.06
5.73/2.70
5.55/2.69
6.08/3.23
5.83/3.06
6.43/3.74
6.22/3.62
6.73/4.20
6.35/3.90
6.35/3.98
5.78/3.49
5.82/3.60
5.10/2.96
5.23/3.17
4.62/2.62
4.76/2.84
4.79/2.94
5.17/3.39
4.96/3.24
Note: Height of first and second barrier used in the GEF code in MeV. (Continuation of Table
2.)
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Special developments
3.2
Nuclear level densities
Nuclear level densities are another important ingredient of any nuclear model. There exist
several descriptions that differ substantially, in particular in their low-energy characteristics. A recent analysis revealed that many of these descriptions are not consistent with
our present understanding of nuclear properties [43]. The result can be summarised as
follows:
1. Even-odd staggering of the nuclear binding energies proves that pairing correlations
are present in essentially all nuclei at low excitation energies. Therefore, any kind
of level-density formula based on the the so-called Fermi-gas level-density, which is
derived under the independent-particle assumption, is not valid in the low-energy
regime.
2. Since pairing correlations are only stable, if they enhance the nuclear binding, the
binding energies of all nuclei are enhanced with respect to the value obtained in the
independent-particle picture. Therefore, energy-shift parameters of the level-density
description for energies above the regime of pairing correlations must be positive for
all nuclei.
3. From an almost constant-temperature behaviour of measured level densities, high
heat capacities are deduced for nuclei up to energies in the order of 10 MeV. Jumps
in the heat capacity prove that the high heat capacity is caused by the consecutive
creation of internal degrees of freedom by pair breaking, such that the energy per
degree of freedom stays approximately constant as a function of excitation energy
in the regime of pairing correlations.
In the GEF model, a simple and transparent analytical description is used that meets
the above-mentioned requirements. The nuclear level density was modelled by the slightly
simplified constant-temperature description of v. Egidy and Bucurescu [44] at low energies.
1
(15)
ρ = e(E−E0 )/T
T
with
E0 = −n · ∆0
(16)
√
(n = 0, 1, 2 for even-even, odd-A, and odd-odd nuclei, respectively, and ∆0 = 12 MeV/ A).
The temperature parameter T depends on the mass A of the nucleus and the shell effect
δU .
A−2/3
T /MeV =
(17)
0.0597 + 0.00198 · δU/MeV
In the ground-state shape this is the ground-state shell effect. The same formula is applied
in different configurations, e.g. at the fission barrier, where the shell effect is basically
different from the
√ ground-state shell effect, however with a larger pairing-gap parameter
∆0 = 14 MeV/ A.
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The level density was smoothly joined at higher energies with the modified Fermi-gas
description of Ignatyuk et al. [45, 46] for the nuclear-state density:
ω∝
√
√
π
exp(2
ãU )
12ã1/4 U 5/4
(18)
with U = E + Econd + δU (1 − exp(−γE)), γ = 0.55/MeV and the asymptotic level-density
parameter ã = √
(0.078A + 0.115A2/3√
)/MeV. The shift parameter Econd = −2 MeV−n∆0 ,
∆0 = 12 MeV/ A (∆0 = 14 MeV/ A at the barrier) with n = 0, 1, 2, for odd-odd, oddA and even-even nuclei, respectively, as proposed in ref. [43]. δU is the shell correction.
A constant spin-cutoff parameter was used. The matching energy is determined from
the matching condition (continuous level-density values and derivatives of the constanttemperature and the Fermi-gas part). Values slightly below 10 MeV are obtained. The
matching condition also determines a scaling factor for the Fermi-gas part. It is related
to the collective enhancement of the level density.
This picture is valid for the situation for near-symmetric mass splits. The increase of
the even-odd effect in Z yields has no influence on the magnitude of the even-odd effect
in total kinetic energy.
3.3
Empirical fragment shells
Figure 6 gives an overview on the measured mass and nuclear-charge distributions of
fission products from low-energy fission. Fission of target nuclei in the actinide region,
mostly induced by neutrons, shows predominantly asymmetric mass splits. A transition
to symmetric mass splits is seen around mass 258 in spontaneous fission of fusion residues.
Electromagnetic-induced fission of relativistic secondary beams covers the transition from
asymmetric to symmetric fission around mass 226 [48]. A pronounced fine structure close
to symmetry appears in 201 Tl [49] and in 180 Hg [50]. It is difficult to observe low-energy
fission in this mass range. Thus, 201 Tl could only be measured down to 7.3 MeV above the
fission barrier due to its low fissility, which explains the filling of the minimum between
the two peaks. Only 180 Hg was measured at energies close to the barrier after beta decay
of 180 Tl. Considering the measured energy dependence of the structure for 201 Tl [49],
the fission characteristics of these two nuclei are rather similar. Also other nuclei in this
mass region show similar features, which have been attributed to the influence of fragment
shells [51]. Nuclei in this region are not further considered in this report that concentrates
on heavier nuclei with mass numbers A > 200, which are more important for technical
applications.
In the range where asymmetric fission prevails, e.g. from 227 Ra to 256 Fm, the light
and the heavy fission-product components gradually approach each other, see Figure
6. A quantitative analysis revealed that the mean mass of the heavy component stays
approximately constant at about A = 140 [52]. This has been explained by the influence
of a deformed (β ≈ 0.6) fragment shell at N = 88 and the spherical shell at N = 82
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[53], suggesting that the position of the heavy fragment is essentially constant in neutron
number.
More recent data on Z distributions over long isotopic chains [48], however, reveal
very clearly that the position in neutron number varies systematically over more than
7 units, while the position in proton number is approximately constant at Z = 54 (see
Figure 7). The rather short isotopic sequences covered in former experiments did not
show this feature clearly enough and gave the false impression of a constant position in
mass. Up to now it has not been possible to identify the fragment shells, which are behind
the observed position of the heavy fragments in the actinides. Although the position of
the heavy fragment is almost constant at Z ≈ 54, it may be doubted that a proton shell
is at the origin of the asymmetric fission of the actinides, because a proton shell in this
region is in conflict with shell-model calculations [53, 55].
At present we limit ourselves in extracting the positions and the shapes of the fission
valleys of the standard 1, the standard 2 and the super-asymmetric fission channels (in the
nomenclature of Brosa et al. [56] ), which form together the asymmetric fission component.
This is done by a fit to the corresponding structures in the measured mass distributions.
Eventually some shells in the complementary fragment are also assumed. The depths of
Figure 6: Systematics of fission-fragment distributions
Note: General view on the systems for which mass or nuclear-charge distributions have been
measured (from [47]). The distributions are shown for 12 selected systems. Blue circles (blue
crosses): Mass (nuclear-charge) distributions, measured in conventional experiments [49, 50],
and references given in [48]. Green crosses: Nuclear-charge distributions, measured in inverse
kinematics [48].
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Figure 7: Mean neutron and proton number of heavy mass peak
Note: Mean neutron and proton number of the heavy component in asymmetric fission in the
actinide region before the emission of prompt neutrons (from [47]). The values were deduced from
measured mass and nuclear-charge distributions using the GEF model for the correction of charge
polarisation and prompt-neutron emission. Open symbols denote results from conventional
experiments, full symbols refer to an experiment with relativistic projectile fragments of 238 U
[48]. Data points for the same ZCN are connected (see [54] for references of the underlying
experimental data).
the fission valleys are deduced from the relative yields of the fission modes by assuming
that the quantum oscillators in the different fission valleys are coupled, which implies
that their populations in the vicinity of the outer barrier are in thermal equilibrium. The
potential at this elongation is calculated as the sum of the macroscopic potential, which
is a function of the fissionning nucleus, and of the shell effects. The magnitudes of the
shell effects are assumed to be the same for all fissioning systems.
Figure 8 illustrates, how the observed transition from symmetric to asymmetric fission
around 226 Th can be explained by the competition of the macroscopic potential that
favours mass-symmetric splits and the shell effect around Z = 55, even if the shell strength
is assumed to be constant. With increasing size of the system, the position of the shell
assumed to be fixed at Z = 55 is found closer to symmetry, which is favoured by the
macroscopic potential. In radium, the potential is lowest at mass symmetry, favouring
single-humped mass distributions, in thorium, the potential at symmetry and near Z = 55
is about equal, creating triple-humped mass distributions, and in plutonium, the potential
is lowest near Z = 55, favouring double-humped mass distributions.
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Figure 8: Potential energy for mass-asymmetric distortions
Note: Schematic illustration of the potential energy for mass-asymmetric shape distortions on
the fission path, after an idea of M. Itkis et al. [57]. The black curve shows the macroscopic
potential that is minimum at symmetry, while the red curve includes the extra binding due to
an assumed shell appearing at Z=55 in the heavy fragment. This is not necessarily a proton
shell (see text for details).
3.4
Charge polarisation
Besides the mass number, also the numbers of protons and neutrons in a fission fragment
can vary independently. In mass-asymmetric fission, the mean Z/A value is found to be
smaller on average in the heavy fragment. The cluster plot of a calculation with the GEF
code, shown in Figure 9, demonstrates this finding on a nuclear chart. Figure 9 shows
the yields of the fission fragments of the system 235 U(nth ,f) before emission of prompt
neutrons. The model reproduces the measured mass-dependent prompt-neutron yields
of 237 Np(n,f) [58] at different incident neutron energies and yields of the light fissionfragments after prompt-neutron emission of 235 U(nth ,f) [59]. Therefore, this calculation is
expected to be quite realistic.
Figure 10 illustrates, how the Z/A degree of freedom is treated in the GEF model.
First, a calculation minimises the energy of the system near scission with respect to the
deformations and the charge densities (Z/A ratio) of the two fragments without considering structural effects. The macroscopic binding energies of the two fragments and the
Coulomb repulsion between the fragments at a tip distance of 1 fm are considered. The
neutron loss by emission of prompt neutrons that is consistent with the available data on
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Figure 9: Nuclide distribution on the N − Z plane
Note: Calculated nuclide distribution produced in thermal-neutron-induced fission of 235 U before
the emission of prompt neutrons. The size of the clusters represents the yield in a logarithmic
scale. The dashed line marks the nuclei with the same N/Z ratio as the fissioning nucleus 236 U.
In addition, the position of the doubly magic 132 Sn and an assumed neutron shell at N = 90
are marked. See text for details.
mass-dependent prompt-neutron yields would not be sufficient to match the Zmean −ZU CD
values of the measured post-neutron nuclide distribution. In order to be consistent with
the systematics of mass-dependent prompt-neutron yields, a charge polarisation of 0.32
units in Zmean − ZU CD of the pre-neutron nuclide distribution, essentially constant over
the whole mass range, before the prompt-neutron emission, must be assumed. Since it is
further assumed that the fission process of all actinides is caused by essentially the same
fragment shells, this polarisation is expected to be the same in the asymmetric fission
channels for all systems.
The fine structure in the curves in Figure 10 results from the even-odd fluctuations in
the fission-fragment yields as a function of atomic number Z and neutron number N .
Figure 9 illustrates the origin of this charge polarisation. The additional binding
energy of spherical nuclei in the S1 fission channel in the vicinity of the N = 82 and the
Z = 50 shells increases when approaching the doubly magic 132 Sn. This explains, why the
fragments in the S1 fission channel tend to be more neutron-rich than expected from the
optimisation of the macroscopic energy. The charge polarisation of the fragments with
deformed shape in the S2 fission channel is explained by the force caused by a shell around
N = 90. This force can be decomposed in a force towards mass asymmetry and a force
towards higher N/Z values at constant mass as illustrated in Figure 9. Since the curvature
of the binding energy against charge polarisation is much larger than the curvature of
the macroscopic potential for mass-asymmetric distortions, the displacement of the mass
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peak from symmetry (∆A = 140-118 = 22) is much larger than the displacement in charge
polarisation (0.32 units).
Figure 10: Mean nuclear charge for fixed fragment mass
Note: Mean nuclear charge of isobaric chains for different cases for the system 235 U(nth ,f).
Dashed line: UCD value. Full line: Macroscopic value at scission. Open symbols: Values
before prompt-neutron emission as a function of pre-neutron mass. Full symbols: Values after
prompt-neutron emission as a function of post-neutron mass.
This reasoning indicates that the S2 fission channel is mainly caused by a deformed
neutron shell, because a proton shell in the heavy fragment would induce a charge polarisation in the opposite direction. Thus, the finding of an almost constant position in
atomic number of the asymmetric fission component in the actinides cannot be attributed
to a proton shell. It must rather be considered as the result of other influences, e.g. of
the competition with the macroscopic potential or additional shells in the light fragment.
3.5
Quantum oscillators of normal modes
There is a long tradition in applying the statistical model to nuclear fission [60, 61, 62, 53].
However, it is well known [63] that the statistical model, applied to the scission-point
configuration, is unable of explaining the variances of the mass and energy distributions
and their dependence on the compound-nucleus fissility parameter. Studies of Adeev
and Pashkevich [14] suggest that dynamical effects due to the influence of inertia and
dissipation can be approximated by considering the properties of the system at an earlier
time. That means that the statistical model may give reasonable results if it is applied
to a configuration that depends on the typical time constant of the collective coordinate
considered.
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The potential U in the vicinity of a minimum as a function of a collective coordinate
q is approximated by a parabola:
1
U = Cq 2
(19)
2
Thus, the motion along the collective coordinate q corresponds to an excited state of
an harmonic quantum oscillator. In an excited nucleus, there is an exchange of energy
between the specific collective and all the other nuclear degrees of freedom that may be
considered as a heat bath. In thermal equilibrium, the properties of the heat bath (e.g.
state density and temperature T ) determine the probability distribution of excited states
of the harmonic oscillator considered. The probability distribution along the coordinate
q is the sum of the contributions from different excited states of the oscillator:
P (q) =
iX
max
0
Wi |φ(q)|2 ,
(20)
where Wi is the population probability of the state i of the oscillator with excitation
energy Ei = i · h̄ω. The upper limit imax is given by the available energy of the system.
If the temperature of the heat bath does not depend on the energy of the nucleus and if
the energy of the nucleus is appreciably higher than the temperature of the heat bath,
the solution is a Gaussian distribution with the variance:
σq2 =
h̄ω
h̄ω
coth( ).
2C
2T
(21)
Two limiting cases are the width of the zero-point motion:
σq2 =
h̄ω
2C
(22)
and the classical limit for T >> h̄ω:
σq2 = T /C
(23)
The evolution of the width of the mass distribution of the symmetric fission channel
at higher excitation energies, where shell effects are essentially washed out, has been the
subject of many experimental and theoretical investigations, see refs. [64, 14, 65, 66].
It was found that the√ width of the mass distribution varies with energy E according
E/a
to the relation σq2 = C that corresponds to the classical limit with the temperature
defined by the Fermi-gas nuclear level density. The values and the variation of the stiffness
parameter C as a function of Z 2 /A agree with theoretical estimations of the stiffness for
mass-asymmetric distortions for a configuration between saddle and scission [66].
A more refined consideration is needed for understanding the mass distribution of a
fission channel in asymmetric fission at energies little above the fission barrier. Here,
the fission valley is formed by a shell effect, which also influences the level density. The
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Figure 11: Widths of the asymmetric fission channels
Note: Standard deviation of the mass distribution of the asymmetric fission channels (standard
1 and standard 2) in the fission of 237 Np(n,f) as a function of the neutron energy En (lower
scale) and the excitation energy above the outer barrier (upper scale). The measured data [67]
(symbols) are compared with the calculated widths of the corresponding quantum oscillators
(dotted lines).
restoring force F (q) of the corresponding oscillator is defined by the variation of the
entropy S:
dS
F (q) = T
,
(24)
dq
and the effective potential U (q) is obtained by integration:
U (q) =
Z
F (q)dq.
(25)
The stiffness C in the vicinity of the potential minimum is given by the second derivative:
C = d2 U/dq 2 .
(26)
In Figure 11, the measured increase of the standard deviation of the mass distribution
of the asymmetric fission channels (standard 1 and standard 2) [67] is compared with the
result of a numerical calculation on the basis of the level-density description presented in
Section 3.2.
The situation is again different for the charge-polarisation degree of freedom. Here,
the potential is dominated by the macroscopic contribution. In addition, the zero-point
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energy is so high that the quantum oscillator is not excited in low-energy fission [68, 69].
Therefore, the width of the charge polarization is essentially a constant value. That does
not mean that the observed width of the isotopic or isobaric distributions is a constant,
because it is broadened by neutron evaporation.
In the GEF code, the widths of the corresponding observables are described with
analytical expressions that represent the physics ideas mentioned above.
3.6
Fission-fragment angular momentum
The empirical data on fission-fragment angular momenta mostly rely on the relative yields
of fission fragments in their isomeric states and on the multiplicity of prompt gamma rays.
Methods for deducing the original angular-momentum distributions from these data rely
on the modelling of the statistical decay of the excited fragments by neutron emission
and gamma de-excitation. The analysis of isomeric yields needs to estimate the angular
momentum carried away by the neutrons and the gammas before reaching the isomer
[70, 71]. The analysis of the gamma multiplicity relies on the distinction of statistical
(dipole) and rotational or vibrational (quadrupole) radiation [72].
The mechanism that is responsible for creating the angular momenta of the fission
fragments has long been controversially discussed. The thermal excitation of angularmomentum-bearing modes [64, 73, 74] within the statistical model requires temperatures
as high as 2 or 3 MeV [75] that might only be possible if strong coupling between collective
degrees of freedom and weak coupling to intrinsic degrees of freedom is assumed. This is
in contradiction to large single-particle excitations found in near magic nuclei [76]. The
pumping of fragment angular momenta by the zero-point motion of these modes has been
found to explain the measured values [73, 77], however with the exception of spherical
fragments in the vicinity of the doubly magic 132 Sn. Also, the torque by electrostatic
repulsion between deformed fragments at scission has been considered [78, 73].
Recently, Kadmensky came up with an idea that seems to solve these problems: He
pointed out that the assumption, often implicitly made, that the orbital angular momentum of the fission fragments is essentially zero, is in conflict with the uncertainty
principle [79]. He assumes that the fluctuation of the orbital angular momentum according to the quantum-mechanical uncertainty is the true principal origin of the fissionfragment angular momenta. The orbital angular momentum is compensated by the fragment angular momenta, eventually also with single-particle excitations in spherical nuclei.
The angular momentum J1 + J2 is shared between the two fragments according to the
ratio of their momenta of inertia I1 and I2 in order to minimise the energy expense
Erot = J12 /(2I1 ) + J22 /(2I2 ).
Since Kadmensky did not derive a quantitative formulation of his idea, the following
semi-empirical description for the spin distribution of one fragment is used:
N (J) = (2J + 1) exp(−
J(J + 1) 2
)
2b
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39
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√
The cut-off parameter b is related to the r.m.s. spin value by Jrms = b/ 2 that is given
by:
q
Jrms = 2Ief f Tef f
(28)
The effective temperature
q
2 + T2
Tnuc
zpm
Tef f =
(29)
considers in a simple way the effect of the zero-point motion of the orbital angular momentum by the parameter Tzpm and the effect of thermal excitations by the nuclear
temperature Tnuc , defined by the energy-dependent nuclear level density.
The reduction of Ief f in the pairing regime:
Ief f = Irigid · (1 − 0.8 · exp(−0.693Eexc /(5 MeV))),
(30)
of the rigid-body momentum of inertia of a fragment with mass number Af with deformation α at scission [80]:
5/3
Irigid = (1.16/fm)2 · Af /(103.8415 MeV/h̄2 ) · (1 + 1/2 · α + 9/7 · α2 ).
(31)
is considered.
The initial spin distribution of the fissioning nucleus is included by considering the
r.m.s. value JCN in a classical approximation:
Jrms =
q
2
2Ief f Tef f + JCN
(32)
In spontaneous fission, JCN is the ground-state spin J0 . In induced fission, it is the
value of the entrance channel. Thus, in neutron-induced fission, the influence of the spin
and the orbital angular momentum of the incident neutron (centre-of-mass energy Ecm )
and the spin of the target nucleus J0 is given by:
JCN =
r
q
J02 + (1/2)2 + (0.1699A1/3 Ecm /MeV)2
(33)
Finally, the observed enhancement of the angular momenta of odd-Z fission fragments4
2/3
is considered by increasing Jrms by the amount Fodd · Af , which depends on the fragment
mass Af . This description has two parameters: Tzpm = 0.8 MeV is adjusted to the
magnitude of measured fission-fragment angular momenta, and Fodd = 0.0148 is deduced
from ref. [81].
For calculating the isomeric yields, once the angular-momentum population of the
fragments is given, the de-excitation cascade by neutron evaporation and gamma emission
should be calculated with full knowledge of the spectroscopic properties and the decay
paths of all fission fragments. Since this knowledge is not available, the GEF model applies
4
We would also expect an even-odd staggering of the fragment angular momentum in neutron number.
However, this effect is not easily observable, because it is washed out by the fluctuations in the promptneutron emission.
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a simple description, following the ideas of refs. [70, 71]. In the angular-momentum
range considered, the emission of neutrons and statistical photons does not modify the
angular-momentum distribution on the average. Therefore, the initial spin distribution
of the fragments is assumed to be preserved until the entrance line is reached, where E2
radiation takes over. These transitions are assumed to follow essentially the yrast line
towards the nuclear ground state. A certain isomeric state, or eventually the ground
state, is assumed to be fed by the angular-momentum range on the entrance line between
the angular momentum of the state considered and the next-higher isomeric state, or
above the highest isomeric state, respectively. This procedure substantially differs from
the often used descriptions of Madland and England [82] and of Rudstam [83]. Madland
and England do not consider the preferred direction of the E2 transitions towards lowerlying states. Rudstam emphasises the change of the fragment spin due to E1 radiation,
which is in conflict with refs. [70, 71], and, in our opinion he overestimates the possibility
that neutron evaporation inhibits the population of high-energy isomers, because most
fission fragments have initial excitation energies sufficiently above the yrast line in the
angular-momentum range considered.
3.7
Energetics of the fission process
In low-energy fission, the available energy, consisting of the Q value of the reaction plus
the initial excitation energy of the fissioning nucleus, ends up either in the total kinetic
energy (TKE) or the total excitation energy (TXE) of the fragments. Moreover, the
TXE is divided between the two fission fragments. In the GEF code, these values are not
parametrised directly. In accordance with the general character of the model, the exchange
of the available energy between the different degrees of freedom of the fissioning system
is described along the fission path. This way, a consistent and complete description of all
phenomena is obtained that depend on the energetics of the fission process. A schematic
presentation of the evolution of the fissioning system is shown in Figure 12.
3.7.1
From saddle to scission
It is assumed that the energy available above the outer saddle (initial excitation energy
of the fissioning nucleus minus the height of the outer fission barrier) is thermalised [11].
This implies that the available energy above the fission barrier is equally shared between
the different degrees of freedom. Most of the available states are intrinsic excitations.
Thus, most of the excitation energy available above the outer saddle is stored in intrinsic
excitations. The intrinsic excitation energy grows on the way from saddle to scission,
because part of the potential-energy release is dissipated into intrinsic excitations. Twocentre shell-model calculations documented in Figure 12 of ref. [12] show that there
are many level crossings on the first section behind the outer saddle. Afterwards, the
single-particle levels change only little. Level crossings lead to intrinsic excitations [85].
Consequently, the additional excitation energy is mostly dissipated in the vicinity of the
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Figure 12: Energetics of the fission process
Note: Schematic presentation of the different energies appearing in the fission process (from
[84]). The vertical dotted line indicates the scission point. The inset illustrates that the energy
release due to the decreasing potential energy is partly dissipated into excitations of collective
normal modes and intrinsic excitations. The remaining part feeds the pre-scission kinetic energy.
The main figure demonstrates that the excitation energy of the fragments still increases right
after scission, because the excess surface energy of the deformed fragments with respect to
their ground states becomes available. Later also the collective excitations are damped into the
intrinsic degrees of freedom. The figure represents the fission of 236 U with an initial excitation
energy equal to the fission-barrier height.
outer saddle. The dissipated energy increases roughly with the mass of the fissioning
nucleus, because the fission barrier is located at smaller deformations, and the range
with a high number of level crossings is extended. Since both the deformation at the
macroscopic barrier and the gain of potential energy from saddle to scission are related
with the Coulomb parameter Z 2 /A1/3 , the amount of dissipated energy from the outer
saddle to scission is assumed to be a constant fraction of the calculated macroscopic
potential energy gain from saddle to scission [86]. Additional intrinsic excitation may
appear at neck rupture.
Theoretical investigations of the gradual transition from the mono-nuclear regime to
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the di-nuclear system concerning shell effects [12], pairing correlations [87] and congruence
energy [88] show that the properties of the individual fission fragments are already well
established in the vicinity of the outer saddle. Therefore, close to the outer saddle the
fissioning system consists of two well-defined nuclei in contact through the neck and a total
amount of excitation energy Etot that is equal to the intrinsic excitation energy above the
outer saddle plus the energy acquired by dissipation on the first section behind the outer
saddle. Intrinsic excitations are expected to be homogeneously distributed within the
nuclear volume. This is likely to hold also in the transition from a mono-nuclear to a
di-nuclear system that takes place very rapidly near the outer saddle [12]. Consequently,
it is assumed that Etot is initially shared among the fragments according to the ratio of
their masses.
We assume that the system formed by the two nuclei in contact then evolves to a
state of statistical equilibrium, the macro-state of maximum entropy, where all the available micro-states have equal probability [89]. This implies that the intrinsic excitation
energy will be distributed among the two nascent fragments according to the probability
distribution of the available microstates which is given by the total nuclear level density.5
Thus, the distribution of excitation energy E1 of one fragment is given by the statistical
weight of the states with a certain division of excitation energy between the fragments:
dN
∝ ρ1 (E1 ) · ρ2 (Etot − E1 )
dE1
(34)
Note that ρ1 and ρ2 are the level densities of the fragments in their shape at scission,
not in their ground-state shape. The remaining energy Etot − E1 is taken by the other
fragment.
In the regime of pairing correlations, where the level density was found to grow almost
exponentially with increasing excitation energy [88-95] energy sorting will take place,
and the light fragment will transfer essentially all its excitation energy to the heavy one
[98, 99]. At higher energies, in the independent-particle regime where pairing correlations
die out, there is a gradual transition to a division closer to the ratio of the fragment
masses according to the validity of the Fermi-gas level density.
The phenomenon of energy sorting explains in a straightforward and natural way
the finding of [58] demonstrated in Figure 13 that the additional energy introduced in
neutron-induced fission of 237 Np raises the neutron multiplicities in the heavy fragment,
only. A similar result was reported for the system 235 U(n,f) [100], but data of this kind
with good quality are scarce.
Part of the energy gain from saddle to scission may also be transferred to collective
modes perpendicular to the fission direction (normal modes [101]). These excitations
correspond to correlated motions of the whole system. The division between the fragments
depends on the kind of collective motion. As an approximation, it is assumed that the
collective excitation energy is equally shared between the fragments.
5
The degeneracy of magnetic sub-states is not considered, because it contributes very little to the
variation of the density of states as a function of excitation energy.
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Figure 13: Variation of the prompt-neutron yield with excitation energy
Note: Prompt-neutron multiplicity as a function of the pre-neutron fragment mass for the system
237 Np(n,f) for E = 0.8 MeV and 5.55 MeV [58]. (Figure from [84].)
n
3.7.2
Fully accelerated fragments
There is widespread agreement that the saw-tooth shape of the prompt- neutron yields
(see Figure 13) is caused by the deformation energies of the nascent fragments at scission.
The scission-point model of ref. [53] attributes it to the influence of fragment shells,
the random-neck-rupture model [56] links it to the location of the rupture, and also
microscopic calculations predict large deformation energies of the fragments near scission
[102]. Large even-odd effects in the fragment Z distributions indicate that the intrinsic
excitation energy at scission is generally much too low to account for the variation of the
prompt-neutron yield by several units over the different fragments.
In the scission-point model of ref. [53], the deformation at scission is determined by
minimising the potential energy for fixed tip distance. An alternative condition would be
to require a fixed distance of the centres of mass of the two nascent fragments. The validity
of one or the other case depends on the magnitude of dissipation [103]. The first case is
valid in the case of strong dissipation, because the relative velocity of the fragments is
slowed down by an attractive force which acts on the nascent fragments through the neck.
The second case is valid for weak dissipation, where the relative velocity of the fragments
reflects the action of the long-range Coulomb force between the nascent fragments. The
magnitude of dissipation in low-energy fission in the regime of strong pairing correlations is
a delicate problem [85]. The GEF model follows the idea of ref. [53]. In this scenario, the
macroscopic forces favour fragments that are strongly deformed (β ≈ 0.5 to 0.6). Thus,
shell effects at these large deformations are favoured, while e.g. the influence of the 132 Sn
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spherical shell is weakened by the macroscopic forces. According to ref. [53], this explains
the weak relative yield of the standard 1 fission channel corresponding to spherical heavy
fragments in the vicinity of 132 Sn, while the bulk of the yield of the asymmetric component
is provided by the standard 2 fission channel with appreciably more strongly deformed
heavy fragments.
In the GEF model, the mean deformation and the width of the different fragment
shells that correspond to the different fission valleys are determined empirically by the
deformation distribution of the light and the heavy fragment that is consistent with the
observed prompt-neutron multiplicity distribution.
After fixing the intrinsic and the collective excitation energy at scission as well as the
fragment deformation energy, the total kinetic energy is determined by energy conservation for a given A and Z split that defines the fission Q value.
The calculated two-dimensional A - Ekin distribution and the TKE distribution for
the system 235 U(nth ,f) are shown in Figures 14 and 15. In addition, Figure 15 demonstrates that the numerical result of the GEF model is not well represented by a normal
distribution. Deviations of measured TKE distributions from a normal distribution have
already been described by Brosa et al. [56] and Zhdanov et al. [104].
Figure 14: A − Ekin distribution
Note: Calculated 2-dimensional A - Ekin distribution for the system
scale gives the number of events per bin.
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th ,f).
The colour
45
Special developments
Figure 15: TKE distribution
Note: Calculated post-neutron TKE distribution for the system
smooth curve shows a fit with a normal distribution.
3.8
235 U(n
th ,f)
(histogram). The
Even-odd effects
Several quantities show a systematic staggering for fragments with even or odd number
of protons. The most striking effect is the enhanced production of even-Z fragments,
for which values up to 40% in the thermal-neutron-induced fission of 232 Th have been
found. But also the yields of even-N fragments are found to be systematically higher.
Moreover, the total kinetic energies of fragments in even-Z charge splits from even-Z
fissioning systems were found to be enhanced.
3.8.1
Z distribution
The enhanced production of even-Z fragments is the most direct observation, because
the number of protons in the fragments is generally not changed after scission since the
probability for proton evaporation from the neutron-rich fission fragments is very low.
In the quasi-particle picture, the ground state is the only state in an even-even nucleus
that is systematically lower than the states in an odd-odd nuclei on an absolute energy
scale. The case of an odd-mass nucleus is in between. Therefore, we attribute the observed
even-odd effect in fission fragment Z distributions to the population of the ground state or
eventually some collective states in the vicinity of the ground state of even-Z fragments.
It seems straightforward to attribute the observed enhanced production of even-Z light
fragments [105] to the energy-sorting mechanism [106] that explained already the differential behaviour of the prompt-neutron yields [98]. If the time until scission is sufficiently
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long for the energy sorting to be accomplished, the system can still gain an additional
amount of entropy by predominantly producing even-even light fragments. Compared to
the production of odd-odd light fragments, the excitation energy of the heavy fragment
increases by two times the pairing gap, and its entropy increases due to the increasing
number of available states.
The quantitative calculation of the even-odd effect is based on the assumption that the
distribution of excited states in the two fragments at scission is in statistical equilibrium.
This means that each state of the fissioning system is populated with the same probability.
For an even-even fissioning nucleus, the number of configurations with Z1 even at fixed
total reduced energy Utot is given by:
NZee1 =e (Z1 )
UtotZ+2∆2
=
ρ1 (U1 )(ee) ρ2 (Utot − U1 )(ee) dU1 +
(35)
−2∆1
Utot
Z +∆2
ρ1 (U1 )(eo) ρ2 (Utot − U1 )(eo) dU1
−∆1
where ρi (Ui )(ee) and ρi (Ui )(eo) are the level densities of representative even-even and evenodd nuclei, respectively, with mass close to A1 or A2 . The reduced energy U is shifted
with respect to the energy E above the nuclear ground state: U = E − n∆, n = 0, 1, 2
for odd-odd, odd-mass, and even-even nuclei, respectively.
The number of configurations with Z1 odd for an even-even fissioning nucleus is:
NZee1 =o (Z1 )
=
Utot
Z −∆2
ρ1 (U1 )(oe) ρ2 (Utot − U1 )(oe) dU1 +
(36)
−∆1
U
Ztot
0
ρ1 (U1 )(oo) ρ2 (Utot − U1 )(oo) dU1
where ρi (Ui )(oe) and ρi (Ui )(oo) are the level densities of representative odd-even and oddodd nuclei, respectively, with mass close to A1 or A2 . The yield for even-Z1 nuclei is
ee
ee
(Z1 ) = NZee1 =e (Z1 ) + NZee1 =o (Z1 ). Similar equa/(Z1 ) with Ntot
YZee1 =e (Z1 ) = NZee1 =e (Z1 )/Ntot
tions hold for odd-even, even-odd and odd-odd fissioning systems. The total available
reduced intrinsic excitation energy Utot is assumed to be a fraction of the potential-energy
difference from saddle to scission. Thus, it increases with the Coulomb parameter Z 2 /A1/3
of the fissioning nucleus.
This approach reproduces the observed salient features of the even-odd effect [105]:
(i) The mean even-odd effect (ΣYZ=e − ΣYZ=o ) / (ΣYZ=e + ΣYZ=o ) decreases with the
Coulomb parameter and with increasing intitial excitation energy. (ii) The local even-odd
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Special developments
effect
δp (Z + 3/2) = 1/8(−1)Z+1 (ln Y (Z + 3) − ln Y (Z) − 3[ln Y (Z + 2) − ln Y (Z + 1)])
increases towards mass asymmetry. (iii) The local even-odd effect for odd-Z fissioning
nuclei is zero at mass symmetry and approaches the value of even-Z nuclei for large mass
asymmetry.
In the GEF code, an analytical function that parametrises the result of this approach
with some adjustment to the measured values is used.
3.8.2
N distribution
In the GEF model it is assumed that the even-odd effect in the yields of the fission
fragments as a function of the atomic number Z is also present in the yields as a function
of the number of neutrons N . However, this structure cannot easily be observed. It is
washed away by prompt-neutron emission, and another even-odd structure is established.
This structure is generated by the influence of the neutron separation energy on the last
stages of the evaporation process, which has also been observed in the cross-section of
projectile fragments in reactions at relativistic energies [107, 108].
An interesting feature of the even-odd staggering of the fission-product yields in neutron number is that the structure generated by the evaporation process is not sensitive
to the excitation energy of the fragments: The structure will remain unchanged with
increasing excitation energy. That is in contrast to the even-odd staggering in atomic
number.
3.8.3
Total kinetic energy
The even-odd effect in the Z distribution of the fission-fragment yields at constant TKE
(or at constant kinetic energy of the light fragment) increases towards higher energy and
decreases towards lower energy. This finding can also be expressed in a different way: The
total-kinetic-energy (TKE) distributions of even-Z elements are shifted to higher values
with respect to their odd-Z neighbours. The magnitude of this shift for thermal-neutroninduced fission of 229 Th, 233 U, 235 U, 239 Pu, 241 Pu, 245 Cm, and 249 Cf is correlated with
the global even-odd effect in the Z yields. This can be deduced from the slope of the
even-odd effect in the Z yields as a function of the kinetic energy of the light fragment
shown in Figure 13 of [109]. In the GEF model, the even-odd fluctuation of the TKE is
calculated by a simple description following an idea of ref. [59]. It is assumed that two
components contribute to the even-Z yield. One component which contains at least one
broken proton pair and the other where no proton pair is broken. If one proton pair is
broken anywhere between the saddle and the scission point, it is assumed that the two
unpaired protons will be distributed statistically on the two fragments. Therefore, the
even-Z yield component with at least one broken pair is equal in amplitude to the oddZ yield, and the energy distributions are expected to be the same, too. In contrast, the
super-fluid component of the even-Z yield is shifted to higher kinetic energies. The shapes
of the energy distributions of the two components of the even-Z yields are assumed to be
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equal. The shift between the two components is the only free parameter of this description.
The data are well described if the two components are assumed to be shifted by 1.7 MeV.
3.9
Spontaneous fission
Fission from excitation energies above or in the vicinity of the fission barrier and spontaneous fission, starting from the nuclear ground state, are very much related. Therefore,
both processes must be described on a common footing. The potential-energy surface is
the same.6 The most important difference is that the passage across the barrier from the
entrance point to the exit point, where the height of the potential exceeds the available
excitation energy, is only possible by tunnelling in spontaneous fission.
In the GEF model, it is assumed that the exit point is located inside one of the fission
valleys. Therefore, the relative yields of the fission channels are given by the relative
values of the transmission coefficients, corresponding to the different fission channels.
The potential energy along the fission path for 240 Pu is schematically shown in Figure 16.
In addition to the passage over the lowest outer barrier, a second passage over another,
0.5 MeV higher, outer barrier of another fission channel is schematically shown.
By the systematics of spontaneous-fission half lives it is known that a variation of
the binding energy of the fissioning nucleus (e.g. by a variation of the ground-state shell
effect) by 1 MeV changes the fission half life by about 5 orders of magnitude [111, 112]. In
that case, the energy deficit is modified over the whole path from entrance point to exit
point. In the present case, it is only the potential around the outer barrier and beyond
which is modified by the influence of fragment shells. Therefore, the relative population
of the different fission valleys is much less sensitive to the magnitudes of the shell effects
in the different fission valleys than the spontaneous-fission half-life is to the ground-state
shell effect. A simple estimate for this sensitivity is given by the Hill-Wheeler formula,
using the h̄ω value of the outer barrier, which is in the order of 0.7 MeV [38, 113]. (We do
not consider the appreciably larger value found for symmetric fission in ref. [113], which
has a very low yield.) However, Figure 16 illustrates that the sensitivity is expected to be
even much weaker, because the potential towards the second minimum is not affected by
the shell effects that form the different fission valleys. In the GEF model, the potential
energy forming the quantum oscillators in mass asymmetry inside the different fission
valleys is assumed to be independent from the excitation energy of the fissioning nucleus.
The sensitivity of the transmission coefficient to the influence of the shell effect that
forms the fission valley is parametrised with the Hill-Wheeler formula with a h̄ω value
around 2 MeV, which is determined by a fit to the experimental fission-fragment yields
in the different fission channels. In view of the above reasoning, this value that is much
6
Note that the potential is defined as the binding energy of the nucleus in the respective shape at
excitation energy zero. The driving force F in a stochastic process is given by the derivative of the
entropy with respect to the collective coordinate q times the temperature: F = T dS/dq [110], not by the
potential. The concept of an excitation-energy-dependent potential-energy surface is only an effective
way to express the properties of the level density above the potential.
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Special developments
Figure 16: Potential energy on the fission path
Note: Schematic drawing of the potential energy as a function of deformation in fission direction.
The heights of the inner and the outer barrier as well as the depth of the second minimum are
those determined for 240 Pu [38] (full line). The potential for another fission path with a 0.5
MeV higher outer barrier is shown in addition (dashed line).
larger than the value governing the tunnelling through the outer barrier appears to be
reasonable. A theoretical estimation of the difference of the transmission for different
fission channels would require a full dynamical quantum-mechanical calculation of the
problem, which is not available with the necessary precision.
Another aspect in which the initial excitation energy of the fissioning nucleus matters
is the energetics of the fission process. The excitation of the quantum oscillators for
mass-asymmetric distortions is considered by a variation of the effective temperature. As
mentioned in Section 3.5, the fluctuation of the charge polarisation is not expected to
vary, because this mode is not excited in low-energy fission anyhow. A variation of the
intrinsic excitation energy at scission is expected to influence the magnitude of the evenodd effect in Z yields, see Section 3.8. Unfortunately, for spontaneous fission this cannot
be confirmed by experiment, because such kind of data is not available.
The description developed for the case of spontaneous fission is also used when the
initial excitation energy falls below the height of the outer fission barrier of a specific
fission channel. The only necessary modification to be considered is the finite initial
energy of the system above the nuclear ground state.
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3.10
Emission of prompt neutrons and prompt gammas
The de-excitation of the fission fragments after scission, including the acceleration phase,
is obtained within the statistical model. Neutron emission during fragment acceleration
reduces especially the laboratory energies of the first neutrons emitted at short times from
highly excited fragments. As discussed in Section 3.8, it is assumed that both the emission
of neutrons and the emission of E1 gammas do not change the angular momentum on
the average, which seems to be a good approximation in the relevant angular-momentum
range [70]. When the yrast line is reached, the angular momentum is carried away by a
cascade of E2 gammas. A global reduction of the momentum of inertia as a function of the
ground-state shell effect according to ref. [114] was applied. Inverse total neutron crosssections with the optical-model parameters of ref. [115] were used. Gamma competition
at energies above the neutron separation energy was considered. The gamma strength of
the giant dipole resonance (GDR) following the description proposed in [116] was applied.
The modelling of the level density of the fragments is described in Section 3.2.
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Particle-induced fission
4
Particle-induced fission
For practical reasons, it is desirable that the GEF code also provides a description of the
processes that happen in induced fission before the formation of an excited compound
nucleus. The technically most important reaction is the interaction of a neutron with a
heavy nucleus.
Neutrons with energies below a few MeV are absorbed, forming a compound nucleus
with an excitation energy equal to the full centre-of-mass energy. Interactions at higher
energies are characterised by the interaction of the incoming neutron with another nucleon
and successive interactions of the excited nucleons with more and more nucleons of the
heavy nucleus. The single-particle configurations develop by more and more complex
patterns towards an equilibrated compound nucleus. During this process, some of the
excited nucleons may be emitted.
The basic idea for the description of this pre-equilibrium emission is the evolution of the
system towards an increasing number of excited particles and holes (excitons). Particles
are emitted from each of these configurations with a characteristic energy spectrum and
with essentially equal probability [117].
More sophisticated pre-equilibrium models were developed, e.g. refs. [118, 119, 120],
which describe the spectra of the emitted nucleons and eventually also of light clusters
with better quality, often with specific empirical adjustments.
Guided by elaborate calculations [121], the probability of pre-equilibrium neutron
emission (Ppe ) up to Nexciton = 10 and statistical emission (Pstat ) is assumed to follow a
linear dependence as a function of the incident-neutron energy Ein :
Ppe /Pstat = (Ein /(M eV ) − 2)/30.
(37)
The shape of the pre-equilibrium neutron spectrum is essentially given by:
dN/dEn ∼ vn · (Ein − En )Nexciton .
(38)
(vn is the velocity of the emitted neutron.) This simple description represents the influence
of pre-equilibrium emission on the prompt-neutron spectrum and the fission-fragment
yields for a fixed incident-neutron energy rather well.
Another aspect of particle-induced fission is the spin distribution of the entrance channel. This aspect is described in Section 3.6.
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5
Multi-chance fission
If the excitation energy of a heavy nucleus is high enough that the excitation energy
after particle emission falls near or above the fission barrier of the daughter nucleus, the
observed fission events are a mixture from the fission of the mother (first-chance fission)
and of the daughter nucleus (second-chance fission). With increasing excitation energy,
also the fission of the grand-daughter nucleus begins to contribute (third-chance fission),
and so on. For the GEF code, it is important to know the relative contributions of the
different chances.
Modelling of multi-chance fission requires calculating the competition between particle
and gamma decay and fission as a function of excitation energy and angular momentum
for determining the relative contributions of the different nuclei and the corresponding
excitation-energy distributions at fission. Moreover, the variation of the fission characteristics, e.g. the nuclide distributions, needs to be described. Both topics are described in
other sections of this report. Figure 17 shows the calculated relative contributions of the
different fission chances to the fission events in the system 235 U(n,f) as a function of the
energy of the incident neutron.
The excitation energies at fission corresponding to the different fission chances are
shown in Figure 18 for the system 235 U(n,f) at En = 14 MeV.
Figure 17: Probabilities of fission chances
Note: Relative contributions of the different fission chances to the fission events of the system
235 U(n,f) as a function of the energy of the incident neutron. Full line: first-chance fission,
dashed line: second-chance fission, dot-dashed line: third-chance fission.
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Multi-chance fission
Figure 18: Excitation energies at fission
Note: Distribution of excitation energies at fission for the system 235 U(n,f) at En = 14 MeV.
The rightmost peak shows events from first-chance fission (fission of 236 U), the middle curve
corresponds to second-chance fission (fission of 235 U), and the left curve corresponds to thirdchance fission (fission of 234 U).
Detailed information on the characteristics of multi-chance fission, in particular at the
threshold energies where a new fission chance opens, is very scarce, because there exist no
comprehensive high-precision data on nuclide distributions with a fine grid of excitation
energies.
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6
Parameter values
According to the concept of the GEF model, a number of parameters were determined
from a systematic analysis of empirical data. In the following, the parameters that are
relevant for the physics of the model are listed. Some details of predominantly technical
interest are not documented in full detail. They may be searched for in the open source
of the code [54].
6.1
Positions of the fission channels
The mean positions of the shell-stabilised heavy fragments of the different fission channels
in thermal-neutron-induced fission are given by the following empirical relations:
For the S1 channel:
Z 1.3
Z̄S1 = 51.5 + 25 · ( CN − 1.5)
(39)
ACN
For the S2 channel:
Z̄S2 = 53.4 + 21.67 · (
1.3
ZCN
− 1.5)
ACN
(40)
Z̄S3 = 58.0 + 21.67 · (
1.3
ZCN
− 1.5)
ACN
(41)
For the S3 channel:
The exact position of the shell around Z = 42 in the light fragment that enhances the
yield of the S1 channel in fissioning nuclei around Pu is:
Z̄light = 42.15.
(42)
The shell in the light fragment that enhances the yield of the S3 channel in fissioning
nuclei around Cf has a slightly different position:
Z̄light = 39.7.
(43)
These two values probably refer to the same shell. In addition to the uncertainties
of this analysis, the displacement can be explained by the correlation between particle
number and deformation for deformed shells as discussed in Section 6.4. The spherical
heavy fragment of the S1 channel induces a stronger Coulomb force and thus drives
the light fragment to larger deformation and the shell to larger particle number than the
strongly deformed heavy fragment of the S3 channel. When this shell in the light fragment
meets one of the shells in the heavy fragment, it enhances the yield of the corresponding
fission channel, but it is apparently too weak to shift its position. There are indications,
however, that this shell has an influence on the deformation of the light fragment at
scission, see Section 6.4.
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Parameter values
According to the assumption that fragment shells are behind the structural effects
that form the fission valleys, this is the same fragment shell, which creates the doublehumped fission-fragment mass distributions for several fissioning systems around Z = 80
[49, 51, 50].
The positions of the fission channels in fragment mass vary with increasing excitation
energy. They are determined by maximising the level density in the mass-asymmetry
degree of freedom, considering the macroscopic potential in mass-asymmetry and the
shell effects.
6.2
Widths of the fission channels
The shape of the potential for mass-asymmetric distortions at the moment of freeze-out
of this degree of freedom is the sum of the macroscopic and the microscopic contribution.
All these contributions are approximated by parabolas U = U0 + c · (Z − Z0 )2 in the
vicinity of their minima, except for the S2 fission channel, where the potential has a more
complex shape. The values of the stiffness coefficients c are listed in Table 4.
Table 4: Stiffness coefficients
macroscopic
systematics [15]
S1
0.30
S3
0.076
Z ≈ 42
0.28
Note: Stiffness coefficients of the different contributions to the potential for mass-asymmetric
distortions at freeze out of this degree of freedom. The stiffness of the macroscopic potential
depends on the system. It is taken from ref. [15].
The shell that forms the S2 channel is parametrised as a rectangular distribution in
particle number with a width of ∆Z = 5.6. The borders are smoothed by a parabolic
shape with c = 0.174 at the lower side and with c = 0.095 at the upper side. This
is technically performed by convoluting the rectangular distribution with two Gaussian
distributions with different width around the two borders of the rectangle. This kind of
shape is consistent with the general feature of deformed shells obtained from shell-model
calculations [52,54], which show an extended valley in the 2-dimensional plane of particle
number and deformation that starts at a specific particle number at small deformation and
extends to a larger particle number at large deformation with a rather constant amount
of additional binding over the whole range.
6.3
Strength of the fragment shells
The strengths of the fragment shells are listed in Table 5. These values refer to the
configuration, where the population of the fission valleys is determined. This is assumed
to be the case little behind the outer fission barrier. The strength of the shell behind the
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S1 fission channel varies as a function of neutron excess, because it is created by both,
the Z = 50 and the N = 82 shells. Thus, its strength decreases if the N/Z ratio of the
fissioning system deviates from the one of the doubly magic 132 Sn.
δUef f = −1.8 · (1 − 4.5 · |82/50 − NCN /ZCN |)
(44)
The maximum value of the effective shell strength δUef f is the sum of the shell strength
δU = −4.6 MeV and the expense ∆Umac = 2.8 MeV to be paid to the macroscopic
potential due to the unfavourable spherical shape.
Table 5: Strengths of the fragments shells near the outer fission barrier.
S2
-4.0 MeV
6.4
S3
-6.0 MeV
Z ≈ 42
-1.3 MeV
Fragment deformation
The shape of the nascent fragments at scission is assumed to be governed by a global
feature that is shown by shell-model calculations [53, 55]. Although the results differ
in their details, the calculations show regular patterns of valleys and ridges extending
from lower particle number and smaller deformation to higher particle number and larger
deformation both for neutron and proton shells. This correlation between shell-stabilised
shape and size of the nucleus is assumed to govern the deformation of the fission fragments
at scission and to explain the saw-tooth behaviour of the prompt-neutron yields.
The deformation of the fragments at scission is approximated by a second-order
spheroid with a tip distance of 1 fm. The deformation parameter β of the heavy fragment
of the S2 fission channel is parametrised as a linear function of the atomic number Zheavy :
βheavy = 0.0275(Zheavy − 48.0).
(45)
The deformation of the light fragment of the S1 and the S2 fission channels is given
by:
βlight = 0.0325(Zlight − 24.5).
(46)
We assume that this is due to a shell, roughly in the region 28 < Z < 50. It was not
possible to deduce the strength of this shell from the fission observables, but it is certainly
weaker than the shells in the heavy fragment, because this shell in the light fragment does
not influence the positions of the S1, S2 and S3 fission channels.
Deviating from this behaviour, the nascent heavy fragment of the S1 channel is assumed to be spherical.
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Parameter values
The deformation parameters of the nascent fragments of the super-long (symmetric)
fission channel were determined by minimising the potential energy (binding energies of
the fragments plus Coulomb interaction potential) at the scission configuration.
The deformation parameter of the heavy fragment of the S3 fission channel is given
by:
βheavy = 0.0275(Zheavy − 48.0) + 0.2.
(47)
The deformation of the light fragment of the S3 fission channel is:
βlight = 0.0325(Zlight − 24.5) − 0.1.
(48)
In all cases, oblate deformation resulting from these formulae was replaced by spherical
shape (β = 0).
For fissioning nuclei around Pu, where the shell around 132 Sn in the heavy fragment
meets the shell near Z = 42 in the light fragment, the deformation of the light fragment
deviates from the above description. The TKE values and the prompt-neutron yields
indicate that the shell near Z = 42 favours less deformed fragments at scission. This
deviation is parametrised accordingly in the GEF code.
After scission, the Coulomb repulsion between the fragments and the condition of
a quasi-fixed tip-distance are not present any more, and, therefore, the fragments snap
to a less deformed shape. In this process, a certain amount of energy is liberated and
adds up to the intrinsic energy of the respective fragment. This energy is assumed to
be dominated by the macroscopic deformation-energy difference given by the liquid-drop
model [122]. Therefore, the contribution due to shell effects is neglected. For the heavy
fragments of the S1 fission channel around 132 Sn this is obviously a realistic assumption,
because these nuclei are nearly spherical at scission and in their ground state. Thus,
the shell effect, which is rather strong, does not change. The shell effects of the other
fragments, typically in the order of a few MeV, are small compared to the variation of
the macroscopic deformation energy, which reaches up to more than 10 MeV.
6.5
Charge polarisation
The charge polarization at scission (related to the deviation of the N/Z ratios of the fragments from the value of the fissioning nucleus) is calculated by minimising the potential
energy of the corresponding scission configuration for a given mass division. In order
to obtain agreement with experimental data, the mean number of protons in the light
(heavy) fragment for a fixed fragment mass is reduced (increased) by 0.32 units, except
for the super-long fission channel. This additional charge polarisation is attributed to the
influence of fragment shells. Because the fragment shells do not depend on the fissioning
system, the polarisation is assumed to be the same for all systems.
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6.6
6.6.1
Energies and temperatures
Temperatures
The width σ of the distribution of the coordinate in a quantum oscillator can be expressed
by an effective temperature TZ that includes the effect of the zero-point motion:
σ=
q
TZ /C
(49)
with
h̄ω
h̄ω
coth( ).
(50)
2
2T
The minimum value of the TZ = h̄ω/2 is not only specific to the collective coordinate,
but, for example for the mass-asymmetry degree of freedom, also to the fission channel
considered.
The temperature parameter for the symmetric fission channel that is created by the
macroscopic potential is given by the parametrisation on the basis of the Fermi-gas level
density of ref. [15] with a minimum value of 0.72 MeV in the constant-temperature
regime. The calculation of the widths of the other fission channels, which are caused by
shell effects, is more complex, see above. They are directly parametrised to reproduce the
empirical values and their variation with excitation energy. The values for the quantum
oscillator for the charge-polarisation degree of freedom are h̄ω = 2 MeV and stiffness
coefficient c = 3.16 MeV (variation of Z for fixed A). The width is dominated by the
zero-point motion.
TZ =
6.6.2
Excitation energy at scission
∗
The total excitation energy at scission consists of three contributions Escission
= EB∗ +
Ediss + Ecoll :
1. The initial excitation energy of the fissioning nucleus minus the height of the outer
fission barrier:
∗
− EB.
(51)
EB∗ = ECN
2. The intrinsic energy acquired through dissipation on the way from the barrier to
scission:
intr
(52)
Ediss = 0.35 · ∆Epot
with
1/3
2
intr
/ACN − 1358) + 11.
/MeV = 0.08 · (ZCN
∆Epot
(53)
This is roughly 35% of the potential-energy gain from saddle to scission given in ref. [86]
with a slight modification.
3. The collective energy acquired through coupling between collective degrees of freedom on the way from the outer barrier to scission:
coll
Ecoll = 0.065 · ∆Epot
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59
Parameter values
with
1/3
2
coll
/ACN − 1390) + 11.
/MeV = 0.08 · (ZCN
∆Epot
(55)
This is roughly 6.5% of the potential-energy gain from saddle to scission given in ref. [86]
with a slight modification.
The dissipated energy at scission Ediss is assumed to fluctuate with a standard deviation of 70 %, not including negative values. The total intrinsic excitation energy at
scission EB∗ + Ediss is subject to energy sorting [98]. The collective energy Ecoll at scission
is shared equally between the fragments.
Note that the excitation-energy-dependent prompt-neutron multiplicities and total
kinetic energies show that due to the lack of suitable transition states below the pairing gap
(2 · ∆ for even-even nuclei and ∆ for odd-mass nuclei), fission at excitation energies above
the barrier proceeds by an effective barrier that is correspondingly higher. Therefore,
Ediss is correspondingly reduced.
6.6.3
Deformation energy
After scission, the deformation of the nascent fragments at scission, which is described
above, invokes an additional excitation energy. It is estimated by the macroscopic deformation energy [122]. The observed fluctuation of the prompt-neutron multiplicity is
mostly explained by the width ∆β = 0.165 of the distribution of the fragment deformation
at scission.
6.6.4
Tunneling
Fission at energies below the outer barrier of a specific fission channel, either in the
ground state or at low excitation energies, is characterised by tunneling and a reduced
value of Ediss . The effective transmission coefficients that determine the populations of
the different fission channels are calculated with the Hill-Wheeler formula. The effective
h̄ω values are expressed by effective temperature parameters Tef f = h̄ω/(2π). Slightly
different values are used for the different fission channels as listed in Table 6. The slightly
larger value for the S1 channel is clearly proven by the data. It is very important for a
good reproduction of the data. It may be connected with a smaller effective mass or with
the more compact configuration at the scission point for this channel.
The reduced value of Ediss is obtained by the formula:
intr
Ediss = 0.35 · ∆Epot
with
1/3
2
intr
∗
/ACN − 1358) + 11 − EB + ECN
/MeV = 0.08 · (ZCN
∆Epot
.
(56)
(57)
Also the value of Ecoll is modified:
coll
Ecoll = 0.065 · ∆Epot
60
(58)
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with
1/3
coll
2
∗
∆Epot
/MeV = 0.1 · (ZCN
/ACN − 1390) + 11 − EB + ECN
.
(59)
Table 6: Effective temperature parameter for tunneling
SL
0.31 MeV
S1
0.342 MeV
S2
0.31 MeV
S3
0.31 MeV
Note: Effective temperatures Tef f for the calculation of the effective transmission coefficients
through the outer fission barrier.
6.7
Concluding remarks
Most of the parameters discussed in this section have a physical meaning and, thus, can
be rather directly compared with results of microscopic theoretical models. Since these
parameters comprise already the knowledge on systematic properties of a large number
of systems, this might give a more valuable constraint than the rather complex body of
direct experimental information.
Altogether, the number of about 50 parameter values from the simultaneous description of a variety of fission observables for almost 100 systems covering from spontaneous
fission to fission at excitation energies of about 100 MeV is astonishingly modest. One
should consider that about the same number of parameter values was used by Wahl [1]
for describing only the fission-fragment yields of only one single system with his empirical
description.
The physical background of the description and its simplicity give confidence that
the GEF model has a good predictive power for nuclei in the neighbourhood of the cases
which were used to constrain the model. Exceptions may exist due to very local structural
effects.
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Uncertainties and covariances
7
Uncertainties and covariances
Experimental data or results of a model calculation are not expected to be precise. Generally they are subject to an uncertainty margin. In both cases, it is important to provide
a realistic estimation of the uncertainty. However, for estimating the uncertainty of an
integral derived quantity that depends on many values, e.g. a whole distribution, the
knowledge of the uncertainty of individual data points is not sufficient.
Often, the variations of different data points are correlated by a contribution from a
common source. A simple case for a common error source for all measured data concerned
is a global normalisation. The uncertainty of the normalisation acts on all data points
in a fully correlated way. In the case of an efficiency curve that is known to be smooth,
the correlation will decrease with the distance between the points of interest. Also, in
the calculated distribution of some observables there exist correlations between different
values, but they have a different origin. If a specific property of the system is changed, this
has an influence on the values of many observables. For example, a decreased dissipation
strength lowers the intrinsic excitation energy at scission and raises the even-odd effect
of the element yields, leading to higher yields for even-Z and lower yields for odd-Z
elements. The fission-fragment yields in the same fission channel are connected by a
positive correlation.
The GEF code provides uncertainties and covariance data for the element yields, the
mass yields and the nuclide yields (depending on Z and A), the latter ones before and after
emission of prompt neutrons. Covariances between any other pair of observables can be
determined by analysing the list-mode output of the GEF code. The covariance between
two observables x and y is determined
number N of calculations with
by performing a large
different sets of parameters~pi = p1 , p2 , · · · , pn . The index i denotes a specific
i
set of parameters. In each set of parameters p~i the values of the different parameters are
chosen randomly from a normal distribution with a central value given by the nominal
parameter value of the model and a standard deviation defined by the uncertainty range
of this parameter. The uncertainty range of a specific parameter of the GEF model was
determined by investigating, how much the parameter value can deviate from the nominal
value, until the agreement with the body of empirical data deteriorates substantially. This
analysis was done with some caution, considering that the comprehensive comparison of
the data with the GEF results gave occasion to distrust some of the experimental or
evaluated data. The determined uncertainties are listed in Table 7. Each parameter is
varied independently from the others. The covariance between the two observables x and
y is defined by:
N
X
(xi − x̄)(yi − ȳ)
Cov(x, y) =
.
(60)
N
i=1
xi and yi are the values of the observables x and y from the individual calculations with
perturbed parameters, x̄ is the mean value of the observable x and ȳ is the mean value of
the observable y of all N calculations. The values of the covariances of a set of observables,
e.g. the yields of the fission-fragment Z distribution form a matrix.
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Table 7: Standard deviations of perturbed parameter values
Quantity
Position of the shell for S1 channel
Position of the shell for S2 channel
Rectangular contribution to the width of S2 channel
Position of the shell for S3 channel
Position of the shell at Z ≈ 42
Shell effect at mass symmetry
Shell effect for S1 channel
Shell effect for S2 channel
Shell effect for S3 channel
Shell at Z ≈ 42
Curvature of shell for S1 channel
Curvature of shell for S2 channel
Curvature of shell for S3 channel
Curvature of shell at Z ≈ 42
(h̄ω)ef f for tunneling of S1 channel
(h̄ω)ef f for tunneling of S2 channel
(h̄ω)ef f for tunneling of S3 channel
(h̄ω)ef f for tunneling of channel at Z ≈ 42
Weakening of the S1 shell with 82/50 − NCN /ZCN
Width of the fragment distribution in N/Z
Charge polarization (Z̄ for fixed A)
σ
0.1
0.1
0.05
0.1
0.1
0.1
0.1
0.1
0.2
0.05
5
5
5
5
3
3
3
3
20
10
0.1
unit
Z units
Z units
Mass units
Z units
Z units
MeV
MeV
MeV
MeV
MeV
%
%
%
%
%
%
%
%
%
%
Z units
Note: Standard deviations of the parameter values used for determining the uncertainties and
the covariances of the GEF results.
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Figure 19: Covariance matrix of fragment mass yields
Note: Covariance matrix of the fission-fragment mass yields after prompt-neutron emission from
the thermal-neutron-induced fission of 239 Pu.
Figure 19 shows the covariance matrix of the mass yields after prompt-neutron emission for the system 239 Pu(nth ,f). The values on the diagonal from the lower-left corner
to the upper-right corner show the largest positive values. They are identical with the
variances of the mass yields. Also, the values from the upper-left corner to the lower-right
corner are positive. These are the covariances between complementary masses. Due to
emission of prompt neutrons, the largest covariance values are slightly smaller and a bit
shifted from the diagonal to the left-lower side. The values of the covariances between the
yields of masses from different fission channels are negative. This is a consequence of the
normalisation of the yields to 200%. The post-neutron mass yields of 239 Pu(nth ,f) including the error bars determined with perturbed-parameter calculations from the GEF code
are compared with the evaluated data from ENDF/B-VII in Figure 20. The estimated
uncertainties of the evaluated data can be seen in Figures 41 and 42.
Assuming that the model is realistic, the covariances between different observables of
a model calculation provide valuable information on the inherent relations between the
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Figure 20: Uncertainties of mass yields from perturbed-parameter calculations
Note: Mass yields after prompt-neutron emission from the thermal-neutron-induced fission of
239 Pu . The GEF result (red full points) with error bars is compared with the evaluated data
from ENDF/B VII (black crosses).
different observables imparted by the physics of the fission process. This information can
be used as a tool to verify the consistency of experimental data and to make evaluations
more efficient.
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8
Assessment
8.1
8.1.1
Fission probability
Introduction
The description of fission observables above the threshold of multi-chance fission requires
the knowledge of the competition between fission, neutron emission and gamma decay as
a function of excitation energy and angular momentum of the compound nucleus, because
they determine the relative weights of the different chances. Entrance-channel-specific precompound processes must eventually be considered in addition. They are not included
in the present study. Since the GEF code aims for modelling the fission process in a
global way without being locally adjusted to experimental data of specific systems, global
descriptions of the relevant decay widths are required. This ensures that the GEF code
can predict fission observables for systems for which no experimental data are available.
However, this also means that specific nuclear-structure effects can only be considered in
an approximate way.
8.1.2
Formulation of the fission probability
The fission probability is calculated as:
Pf = Γf /(Γf + Γn + Γγ ).
(61)
The gamma-decay width is calculated by the global formula:
5
Γγ = 0.62410−9 · A1.6
CN · Ti M eV
(62)
proposed by Ignatyuk [46]. ACN is the mass number and Ti is the temperature of the
compound nucleus with energy Ei .
The neutron-decay width is calculated by the global formula:
Γn = 0.13 · (ACN − 1)2/3 · Tn2 / exp(< Sn > /Tn )
(63)
proposed in ref. [123], which is valid for an exponential neutron-energy spectrum. Sn is
the neutron separation energy, Tn is the maximum temperature of the daughter nucleus
at the energy Ei − < Sn >. This expression was multiplied by:
1 − exp(−(Ei − < Sn >)/(1.6 · Tn ))
(64)
in order to approximately adapt to the Maxwellian shape of the neutron-energy spectrum.
The use of < Sn >= S2n /2 is another way to consider the shift of the level density by ∆
and 2∆ in odd-mass and even-even nuclei, respectively, with respect to odd-odd nuclei.
Γn is set to zero at energies below the neutron separation energy Sn .
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The calculation of the fission-decay width is based on the following equations proposed
in ref. [124] with a few extensions:
Γf = Frot · Tf /(G · exp(Bm /Tf )).
(65)
Bm is the maximum value of the inner fission barrier BA and the outer barrier BB , Tf
is the temperature of the compound nucleus at the barrier Bm . Frot = exp((Irms /15)2 )
considers the influence of the root-mean square value Irms of the angular-momentum
distribution of the compound nucleus.
G = GA · exp((BA − Bmax )/Tf ) + GB · exp((BB − Bmax )/Tf )
(66)
whereby GA and GB consider the collective enhancement of the level densities on top of the
inner barrier(assuming triaxial shapes) and the outer barrier (assuming mass-asymmetric
shapes) and of tunneling through the corresponding barrier:
q
GA = FA · 0.14/ π/2,
(67)
FA = 1/(1 + exp(−(E − BA )/Tequi ),
(68)
GB = FB /2,
(69)
FB = 1/(1 + exp(−(E − BB )/Tequi ).
(70)
Fatt = exp(0.05(E − BA ))/(1/GA + exp(0.05(E − BA )))
(71)
and
Tequi is related to the values of h̄ωA and h̄ωB at the inner and outer barriers by Tequi =
h̄ω/2π, assuming h̄ωA = h̄ωB = 0.9 MeV.
In order to account for the low level density above Bm at energies below the pairing
gap 2∆ in even-even nuclei, the value of Γf was multiplied at energies in the vicinity of
the barrier Bm by a reduction factor that was deduced from the average behaviour of
measured fission probabilities. The function is shown in Figure 21.
The collective-enhancement factors at the inner and outer barrier with respect to the
daughter nucleus
q after neutron decay that is assumed to have a quadrupole shape (the
inverse of 0.14/ π/2 and 0.5, respectively) are assumed to fade out at higher energies,
where the shape of the fissioning nucleus at scission becomes mass symmetric. They are
multiplied by the attenuation factor:
for the inner barrier and an analogous factor for the outer barrier.
The temperature values were determined as the inverse logarithmic derivative of the
nuclear level density with respect to excitation energy. The nuclear level density both
in the ground-state minimum and at the fission barrier was modelled by the constanttemperature description of v. Egidy and Bucurescu [44] at low energies. The level density
was smoothly joined at higher energies with the modified Fermi-gas description of Ignatyuk et al. [45, 46] for the nuclear-state density:
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Figure 21: Reduction of fission-decay width
Note: Adapted reduction of the fission-decay width around the fission barrier for even-even
nuclei
√
√
π
exp(2
ãU )
(72)
12ã1/4 U 5/4
with U = E +Econd +δU (1−exp(−γE)), γ = 0.55 and the asymptotic level-density √
pa2/3
rameter ã = 0.078A+0.115A . The shift parameter Econd = 2 MeV −n∆0 , ∆0 = 12/ A
with n = 0, 1, 2, for odd-odd, odd-A and even-even nuclei, respectively, as proposed in ref.
[43]. δU is the ground-state shell correction. Because the level density in the low-energy
range is described by the constant-temperature formula, a constant spin-cutoff parameter
was used. The matching energy is determined from the matching condition (continuous
level-density values and derivatives of the constant-temperature and the Fermi-gas part).
Values slightly below 10 MeV are obtained. The matching condition also determines a
scaling factor for the Fermi-gas part. It is related with the collective enhancement of the
level density.
The fission barriers were modelled on the basis of the Thomas-Fermi fission barriers
of Myers and Swiatecki [36], using the topographical theorem of the same authors [20] to
account for the contribution of the ground-state shell effect. Adjustments to measured
barrier values [40] were applied. Details are described in Section 3.
ω∝
8.1.3
Comparison with experimental data
Figures 22 to 30 show a survey on measured fission probabilities in comparison with the
results of the GEF code. The data are taken from the following publications: [125, 126,
127, 128, 129, 130]. Some of the figures show the data from different reactions with
different symbols. (See the original publications for details.)
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Figure 22: Benchmark of fission probabilities, part 1
Note: Comparison of the measured fission probabilities (black symbols) with calculations with
the GEF code (red symbols). The fission barrier and the neutron separation energy used in the
calculations are listed.
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Figure 23: Benchmark of fission probabilities, part 2
Note: Comparison of the measured fission probabilities (black symbols) with calculations with
the GEF code (red symbols).
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Figure 24: Benchmark of fission probabilities, part 3
Note: Comparison of the measured fission probabilities (black symbols) with calculations with
the GEF code (red symbols).
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Figure 25: Benchmark of fission probabilities, part 4
Note: Comparison of the measured fission probabilities (black symbols) with calculations with
the GEF code (red symbols).
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Figure 26: Benchmark of fission probabilities, part 5
Note: Comparison of the measured fission probabilities (black symbols) with calculations with
the GEF code (red symbols).
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Figure 27: Benchmark of fission probabilities, part 6
Note: Comparison of the measured fission probabilities (black symbols) with calculations with
the GEF code (red symbols).
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Figure 28: Benchmark of fission probabilities, part 7
Note: Comparison of the measured fission probabilities (black symbols) with calculations with
the GEF code (red symbols).
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Figure 29: Benchmark of fission probabilities, part 8
Note: Comparison of the measured fission probabilities (black symbols) with calculations with
the GEF code (red symbols).
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Figure 30: Benchmark of fission probabilities, part 9
Note: Comparison of the measured fission probabilities (black symbols) with calculations with
the GEF code (red symbols).
8.1.4
Discussion
The absolute values and the energy dependence of the fission probabilities of most systems
reaching from Pa to Cm are rather well reproduced by the GEF code at energies above
the fission barrier. However, in many cases, fission sets in at too low energies in the
calculation. In a few cases, the measured fission probabilities are considerably lower than
the calculated ones, while the threshold and the energy dependence are rather similar.
The most pronounced cases are 229 Th, 230 Th, 231 Th, 233 Th, and 234 U.
A possible key to the latter problem may be seen in the figures for 231 Pa, 235 Np,
239
Pu, 240 Pu, and 244 Cm, where different sets of measured data exist. In all these cases,
one of the data sets gives appreciably higher values than the other one, and the higher
values agree rather well with the model calculations. For fission probabilities obtained
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with transfer reactions, there may be a background originating from reactions on target
contaminants (e.g. oxygen) or from other parasitic reactions like the breakup of the
projectile (deuteron-breakup in particular). This may explain the differences encountered
between the different groups of experimental data. Thus, this problem might have its
origin in the experimental data at least in some of the cases.
The deviations at the threshold may be attributed to the shortcoming of the model
due to its global description. Specific structural effects at low excitation energies, in
particular structural information on the levels above the fission barrier are not properly
considered. The observed deviations correspond to a shift of the effective threshold in
the order of several 100 keV. Considering that the fission barriers extracted by different
authors for the same nucleus often differ by 0.5 MeV and more [40], the deviations are not
too surprising. The kind of disagreement seen in the figures gives a realistic impression
about the quality of the predictions of the model for cases, where no experimental data
exist.
8.1.5
Conclusion
A global description of the fission probability of the actinides has been derived which
reproduces the experimental data rather well. Discrepancies in the absolute values over
the whole energy range might be caused by a background contribution due to the presence
of light target contaminants in the experiment. The global description of the nuclear level
densities near the ground state and near the fission threshold used in the code can only
give a rather crude approximation of the behaviour of the fission probability near the
fission threshold. This explains the discrepancies in the fission probabilities near the
fission threshold found for several systems. The energy-dependent fission probabilities
are important to calculate the relative weights of the different fission chances at higher
energies.
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8.2
8.2.1
Fission-fragment yields
Introduction
Several hundred different nuclides are produced in the fission of a heavy nucleus. They
essentially contribute to the radioactive inventory of a fission reactor, and they are the
source of most part of the decay heat that incurres in the fuel rods even long time after
the shut-down of a fission reactor. The relative yields of the different nuclides depend on
the fissioning nucleus and on the excitation energy at fission. Moreover, the radioactivedecay properties of the different fission products differ very much. Therefore, a very good
knowledge on the yields of the different fission products is of paramount importance for
the operation of a fission reactor and for the storage of used fuel rods.
New data are required when new generations of fission reactors are developed, e.g.
when fission is induced by neutrons of higher energies, or when eventually other kind of
fuel is used. Reliable model calculations of the fission-product yields are urgently required
which can replace time-consuming and expensive experiments.
The GEF code [131, 54, 132] has been developed with the aim to provide this kind
of information. In the following, the quality and the predictive power of the GEF code
for calculating fission-fragment yields for different fissioning systems and a large range of
energy is assessed.
8.2.2
Experimental techniques
It is worthwhile to have a look on the different most often used experimental techniques
applied to measure fission-product yields, because they determine the nature of the data.
The traditional approach is based on the identification of gamma rays that are characteristic for the radioactive decay of a specific fragment [133]. Fission-product masses
after the emission of prompt neutrons are determined. This technique is able to identify
the emitting nuclide unambiguously, but it requires additional knowledge on the decay
properties, e.g. branching ratios, in order to deduce quantitative yields. Moreover, this
technique is not well suited for measuring the yields of short-lived fission products and
unable to determine the yields of stable nuclides.
The masses of the fission products can also be determined by particle detectors that
measure the energies and/or the velocities of the fission products [134, 135, 136]. These
methods are suited to deduce the masses of the fission products before and after the
emission of prompt neutrons. However, the resolution is not sufficient to determine the
mass unambiguously in most cases.
Unambiguous determination of fission-product masses is achieved by use of the Lohengrin spectrograph [137] at the high-flux reactor of the ILL, Grenoble. Also, the nuclear charge in the light group of the fission products can be determined. However, this
technique is restricted to thermal-neutron-induced fission and a limited choice of target
material.
A novel kind of experiments in inverse kinematics [138, 48, 139] succeeded to determine
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the mass A and the atomic number Z of all fission products unambiguously in Coulomb
fission of short-lived neutron-deficient projectile fragments at relativistic energies. The full
identification of all fission products in A and Z was also achieved in the fission of transfer
products from 238 U projectiles at energies above the Coulomb barrier [140]. Besides the
unprecedented resolution in kinematical measurements, these experiments offer a wide
choice of fissionning systems, not accessible before.
8.2.3
Mass distributions
Fission-fragment mass distributions have a particular importance. First, they are determined with full resolution for a large number of systems by gamma-spectroscopic measurements. The data which are often incomplete are completed with the Wahl systematics
[1]. Moreover, the beta decay which is the predominant decay path follows the mass chain.
Thus, the mass distribution can also be deduced from cumulative yields. Second, the mass
distributions allow estimating the long-term radioactive decay characteristics rather well,
because beta decay that connects nuclei with the same mass number is the predominant
decay path in most cases.
In the following, experimental and evaluated mass distributions in four different energy
classes and from different experimental sources are compared with the result of the GEF
code. Depending on the experimental technique, mass distributions before emission of
prompt neutrons (Apre ) and after emission of prompt neutrons (Apost ) are given. In a few
cases, Aprov , the provisional mass, is shown. It is directly deduced from the ratio of the
kinetic energies E1 and E2 of the fragments, assuming that A1 /A2 = E2 /E1 , and, thus, it
is not corrected for neutron emission.
The calculated individual contributions from the different fission modes are shown in
addition. The comparison is not exhaustive, but it gives a rather complete view on the
variation of the mass distributions from protactinium to rutherfordium. The error bars
represent the uncertainties given in the indicated references (see Table 8).
Although the GEF code is able to produce uncertainties by calculations with perturbed
parameters, they are not shown in order not to overload the figures.
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8.2.3.1 Spontaneous fission: Figures 31 to 38 show the calculated mass distributions
for spontaneous fission in comparison with measured or evaluated data of a number of
systems in linear and in logarithmic scale. Although spontaneous fission is less important for technical applications, these figures are essential for revealing the dependence
of the fission-fragment yields upon excitation energy on the fission path. Note that the
kinematical measurements of pre-neutron masses are subject to a finite resolution and
uncertainties due to the correction for detector response and prompt-neutron emission.
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Figure 31: Mass distributions, spontaneous fission, part 1, linear scale
Note: Evaluated and experimental mass distributions (black symbols) of fission fragments in
comparison with the results of the GEF code (green and blue symbols) in a linear scale. Spontaneous fission, part 1.
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Figure 32: Mass distributions, spontaneous fission, part 1, logarithmic scale
Note: Evaluated and experimental mass distributions (black symbols) of fission fragments in
comparison with the results of the GEF code (green and blue symbols) in a logarithmic scale.
Spontaneous fission, part 1.
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Figure 33: Mass distributions, spontaneous fission, part 2, linear scale
Note: Evaluated and experimental mass distributions (black symbols) of fission fragments in
comparison with the results of the GEF code (green and blue symbols) in a linear scale. Spontaneous fission, part 2.
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Figure 34: Mass distributions, spontaneous fission, part 2, logarithmic scale
Note Evaluated and experimental mass distributions (black symbols) of fission fragments in
comparison with the results of the GEF code (green and blue symbols) in a logarithmic scale.
Spontaneous fission, part 2.
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Figure 35: Mass distributions, spontaneous fission, part 3, linear scale
Note: Evaluated and experimental mass distributions (black symbols) of fission fragments in
comparison with the results of the GEF code (green and blue symbols) in a linear scale. Spontaneous fission, part 3.
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Figure 36: Mass distributions, spontaneous fission, part 3, logarithmic scale
Note: Evaluated and experimental mass distributions (black symbols) of fission fragments in
comparison with the results of the GEF code (green and blue symbols) in a logarithmic scale.
Spontaneous fission, part 3.
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Figure 37: Mass distributions, spontaneous fission, part 4, linear scale
Note: Evaluated and experimental mass distributions (black symbols) of fission fragments in
comparison with the results of the GEF code (green and blue symbols) in a linear scale. Spontaneous fission, part 4.
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Figure 38: Mass distributions, spontaneous fission, part 4, logarithmic scale
Note: Evaluated and experimental mass distributions (black symbols) of fission fragments in
comparison with the results of the GEF code (green and blue symbols) in a logarithmic scale.
Spontaneous fission, part 4.
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8.2.3.2 Fission induced by thermal neutrons: Mass distributions from thermalneutron-induced fission are shown in Figures 39 to 44.
Figure 39: Mass distributions, (nth , f ), part 1, linear scale
Note: Evaluated and experimental mass distributions (black symbols) of fission fragments in
comparison with the results of the GEF code (green and blue symbols) in a linear scale. Thermalneutron-induced fission, part 1.
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Figure 40: Mass distributions, (nth , f ), part 1, logarithmic scale
Note: Evaluated and experimental mass distributions (black symbols) of fission fragments in
comparison with the results of the GEF code (green and blue symbols) in a logarithmic scale.
Thermal-neutron-induced fission, part 1.
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Figure 41: Mass distributions, (nth , f ), part 2, linear scale
Note: Evaluated and experimental mass distributions (black symbols) of fission fragments in
comparison with the results of the GEF code (green and blue symbols) in a linear scale. Thermalneutron-induced fission, part 2.
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Figure 42: Mass distributions, (nth , f ), part 2, logarithmic scale
Note: Evaluated and experimental mass distributions (black symbols) of fission fragments in
comparison with the results of the GEF code (green and blue symbols) in a logarithmic scale.
Thermal-neutron-induced fission, part 2.
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Figure 43: Mass distributions, (nth , f ), part 3, linear scale
Note: Evaluated and experimental mass distributions (black symbols) of fission fragments in
comparison with the results of the GEF code (green and blue symbols) in a linear scale. Thermalneutron-induced fission, part 3.
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Figure 44: Mass distributions, (nth , f ), part 3, logarithmic scale
Note: Evaluated and experimental mass distributions (black symbols) of fission fragments in
comparison with the results of the GEF code (green and blue symbols) in a logarithmic scale.
Thermal-neutron-induced fission, part 3.
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8.2.3.3 Fission induced by fast neutrons: Fast-neutron-induced fission comprises
the energy range of fission neutrons up to a few MeV. Some data refer to well defined
energies of e.g. 0.4 MeV or 2 MeV, some correspond to rather broad energy distributions.
In the case of a strongly energy-dependent fission cross section, the mean energy of the
fissioning nuclei may be rather high. Therefore, in Figures 45 to 8.2.3 the strongly energydependent yield at symmetry may often not be correctly reproduced by the calculations,
which were performed with an incoming-neutron energy of 2 MeV in all cases.
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Figure 45: Mass distributions, (nf ast , f ), part 1, linear scale
Note: Evaluated and experimental mass distributions (black symbols) of fission fragments in
comparison with the results of the GEF code (green and blue symbols) in a linear scale. Fastneutron-induced fission, part 1.
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Figure 46: Mass distributions, (nf ast , f ), part 1, logarithmic scale
Note: Evaluated and experimental mass distributions (black symbols) of fission fragments in
comparison with the results of the GEF code (green and blue symbols) in a logarithmic scale.
Fast-neutron-induced fission, part 1.
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Figure 47: Mass distributions, (nf ast , f ), part 2, linear scale
Note: Evaluated and experimental mass distributions (black symbols) of fission fragments in
comparison with the results of the GEF code (green and blue symbols) in a linear scale. Fastneutron-induced fission, part 2.
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Figure 48: Mass distributions, (nf ast , f ), part 2, logarithmic scale
Note: Evaluated and experimental mass distributions (black symbols) of fission fragments in
comparison with the results of the GEF code (green and blue symbols) in a logarithmic scale.
Fast-neutron-induced fission, part 2.
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Figure 49: Mass distributions, (nf ast , f ), part 3, linear scale
Note: Evaluated and experimental mass distributions (black symbols) of fission fragments in
comparison with the results of the GEF code (green and blue symbols) in a linear scale. Fastneutron-induced fission, part 3.
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Figure 50: Mass distributions, (nf ast , f ), part 3, logarithmic scale
Note: Evaluated and experimental mass distributions (black symbols) of fission fragments in
comparison with the results of the GEF code (green and blue symbols) in a logarithmic scale.
Fast-neutron-induced fission, part 3.
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Figure 51: Mass distributions, (nf ast , f ), part 4, linear scale
Note: Evaluated and experimental mass distributions (black symbols) of fission fragments in
comparison with the results of the GEF code (green and blue symbols) in a linear scale. Fastneutron-induced fission, part 4.
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Figure 52: Mass distributions, (nf ast , f ), part 4, logarithmic scale
Note: Evaluated and experimental mass distributions (black symbols) of fission fragments in
comparison with the results of the GEF code (green and blue symbols) in a logarithmic scale.
Fast-neutron-induced fission, part 4.
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8.2.3.4 Fission induced by 14-MeV neutrons: A few mass distributions were measured for fission induced by 14-MeV neutrons. They are compared with results of the
GEF code in Figures 53 to 56.
Figure 53: Mass distributions, En = 14 MeV, part 1, linear scale
Note: Evaluated and experimental mass distributions (black symbols) of fission fragments in
comparison with the results of the GEF code (green and blue symbols) in a linear scale. 14MeV-neutron-induced fission, part 1.
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Figure 54: Mass distributions, En = 14 MeV, part 1, logarithmic scale
Note: Evaluated and experimental mass distributions (black symbols) of fission fragments in
comparison with the results of the GEF code (green and blue symbols) in a logarithmic scale.
14-MeV-neutron-induced fission, part 1.
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Figure 55: Mass distributions, En = 14 MeV, part 2, linear scale
Note: Evaluated and experimental mass distributions (black symbols) of fission fragments in
comparison with the results of the GEF code (green and blue symbols) in a linear scale. 14MeV-neutron-induced fission, part 2.
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Figure 56: Mass distributions, En = 14 MeV, part 2, logarithmic scale
Note: Evaluated and experimental mass distributions (black symbols) of fission fragments in
comparison with the results of the GEF code (green and blue symbols) in a logarithmic scale.
14-MeV-neutron-induced fission, part 2.
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8.2.4
Deviations
The reduced Chi-squared values of the deviations between GEF results and evaluated data
are given in Tables 8 to 10. There are no Chi-squared values given for distributions of preneutron or provisional masses because of several reasons. In some cases, the uncertainties
are not available. Moreover, the data are disturbed by the finite mass resolution and
possible problems in the corrections for prompt-neutron emission [19]. The tables also
give the references to the sources of the data shown in Figures 31 to 56.
Table 8: Measured and evaluated mass distributions, part 1
System
238
U(sf)
Pu(sf)
240
Pu(sf)
242
Pu(sf)
244
Pu(sf)
244
Cm(sf)
246
Cm(sf)
248
Cm(sf)
250
Cf(sf)
252
Cf(sf)
253
Es(sf)
254
Fm(sf)
256
Fm(sf)
258
Fm(sf)
259
Md(sf)
260
Md(sf)
256
No(sf)
258
No(sf)
238
Measured
quantity
Reference
Apost
Apre
Apre
Apre
Apre
Apost
Apost
Apost
Apost
Apost
Apost
Apost
Apost
Aprov
Aprov
Aprov
Apre
Aprov
[141]
[142]
[142]
[142]
[143]
[141]
[141]
[141]
[141]
[141]
[141]
[141]
[141]
[144]
[144]
[144]
[145]
[144]
reduced
Chisquared
3.2
—
—
—
—
1.4
1.1
7.8
0.9
0.6
3.7
0.4
0.8
—
—
—
—
—
Note: Measured and evaluated mass distributions used for the comparison in Figures 31 to 56,
their nature and their references. The last column gives the sum of the squared deviations of
the GEF results from the evaluated yields divided by the uncertainties of the empirical data per
degrees of freedom (reduced Chi-squared). This value should be around 1 for a good description.
Only yields larger than 0.01% have been considered.
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Table 9: Measured and evaluated mass distributions, part 2
System
259
Lr(sf)
Rf(sf)
262
Rf(sf)
227
Th(nth ,f)
229
Th(nth ,f)
232
U(nth ,f)
233
U(nth ,f)
235
U(nth ,f)
237
Np(nth ,f)
239
Pu(nth ,f)
240
Pu(nth ,f)
241
Pu(nth ,f)
242
Pu(nth ,f)
241
Am(nth ,f)
242
Am(nth ,f)
243
Cm(nth ,f)
245
Cm(nth ,f )
249
Cf(nth ,f)
251
Cf(nth ,f)
254
Es(nth ,f)
255
Fm(nth ,f)
232
Th(n,f), En =fast
231
Pa(n,f), En =fast
233
U(n,f), En =fast
234
U(n,f), En =fast
235
U(n,f), En =fast
236
U(n,f), En =fast
237
U(n,f), En =fast
238
U(n,f), En =fast
237
Np(n,f), En =fast
260
Measured
quantity
Reference
Apost
Aprov
Apre
Apost
Apost
Apost
Apost
Apost
Apost
Apost
Apost
Apost
Apost
Apost
Apost
Apost
Apost
Apost
Apost
Apost
Apost
Apost
Apost
Apost
Apost
Apost
Apost
Apost
Apost
Apost
[146]
[144]
[147]
[141]
[141]
[141]
[141]
[141]
[141]
[141]
[141]
[141]
[141]
[141]
[141]
[141]
[141]
[141]
[141]
[141]
[141]
[141]
[141]
[141]
[141]
[141]
[141]
[141]
[141]
[141]
reduced
Chisquared
—
—
—
420
26
1.6
1.2
7.4
1.1
1.2
0.9
0.7
0.8
0.6
0.9
1.7
0.5
0.7
4.9
9.5
5.6
1.7
1.0
1.0
1.0
1.5
1.2
0.4
1.1
0.5
Note: Table 8 continued
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Table 10: Measured and evaluated mass distributions, part 3
System
Measured
quantity
Reference
Np(n,f), En =fast
Pu(n,f), En =fast
239
Pu(n,f), En =fast
240
Pu(n,f), En =fast
241
Pu(n,f), En =fast
242
Pu(n,f), En =fast
241
Am(n,f), En =fast
243
Am(n,f), En =fast
242
Cm(n,f), En =fast
244
Cm(n,f), En =fast
246
Cm(n,f), En =fast
248
Cm(n,f), En =fast
232
Th(n,f), En =14 MeV
233
U(n,f), En =14 MeV
234
U(n,f), En =14 MeV
235
U(n,f), En =14 MeV
236
U(n,f), En =14 MeV
238
U(n,f), En =14 MeV
237
Np(n,f), En =14 MeV
239
Pu(n,f), En =14 MeV
240
Pu(n,f), En =14 MeV
242
Pu(n,f), En =14 MeV
241
Am(n,f), En =14 MeV
Apost
Apost
Apost
Apost
Apost
Apost
Apost
Apost
Apost
Apost
Apost
Apost
Apost
Apost
Apost
Apost
Apost
Apost
Apost
Apost
Apost
Apost
Apost
[141]
[141]
[141]
[141]
[141]
[141]
[141]
[141]
[141]
[141]
[141]
[141]
[141]
[141]
[141]
[141]
[141]
[141]
[141]
[141]
[141]
[141]
[141]
238
238
reduced
Chisquared
0.5
0.7
0.6
0.6
0.7
0.6
0.7
1.4
1.4
1.0
1.0
1.5
1.7
0.7
0.3
0.5
0.4
0.3
0.6
0.5
0.7
0.5
0.6
Note: Table 8 continued
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8.2.5
Charge polarisation and emission of prompt neutrons
The two fission products are not fully determined by their mass, because the protons and
neutrons of the fissioning nucleus can be distributed in a different portion to the fragments
at scission by charge polarisation. Furthermore, the emission of prompt neutrons from the
excited fragments tends to decrease the neutron excess of the fragments. For calculating
the nuclide distributions after prompt-neutron emission for which the most precise data
are available, both the charge polarisation and the prompt-neutron emission must be
considered. Data on the mass-dependent prompt-neutron multiplicity exist for a few
fissioning systems. They provide the necessary information for disentangling the influence
of charge polarisation and prompt-neutron emission on the neutron excess of the fission
products.
Figure 57 shows the mean values and the standard deviations of the Z distributions for
fixed post-neutron mass. Available experimental data in the light fission-fragment group
of four fissioning systems [148, 59, 149, 150, 151] are compared with the result of the GEF
code. The agreement is generally very good, except for the heaviest nuclei of the system
249
Cf(nth ,f). However, it is not clear, whether this discrepancy can be attributed to a
shortcoming of the model, because the experiment suffered from insufficient Z resolution
in this range.
The mass-dependent mean prompt-neutron multiplicity for 237 Np(n,f) [58] and 252 Cf(sf)
[152] are shown below in the dedicated section. Also here one can observe a good reproduction of the experimental data by the GEF code.
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Figure 57: Charge density of post-neutron fragments
Note: Deviation of the mean fission-product nuclear charge for a fixed mass after prompt-neutron
emission from the UCD value. Experimental data (black symbols) are compared with the results
of the GEF code (red symbols).
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8.2.6
Nuclide distributions
A detailed overview on the nuclide distribution is shown in Figures 58 to 67 for only one
representative system because of the large quantity of this kind of data. In these figures,
the empirical mass-chain yields for the system 235 U(nth ,f) are compared with the result
of the GEF code. In the light fission-product group, the data measured at the Lohengrin
spectrograph have been chosen [59], while the evaluation of A. Wahl [1] was used for the
heavy group. However, only those elements of this evaluation were taken for which at
least two data points were directly derived from experimental data.
In Figures 58 to 62, a logarithmic scale has been chosen, spanning the same range
from 10−4 % to 10%. The error bars of the empirical data represent the uncertainties
given in [59, 1]. The error bars of the GEF results represent the estimated uncertainties
that were obtained by calculations with perturbed parameters. The relevant parameters
were varied within their uncertainty range.
Figure 58: Isobaric Z distributions for
238
U(nth ,f), part 1, logarithmic scale
Note: Post-neutron isobaric element distributions in logarithmic scale. The experimental data
[59] (black symbols) are compared with the results of the GEF code (red symbols). The mass
numbers are specified in the figures.
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Figure 59: Isobaric Z distributions for
238
U(nth ,f), part 2, logarithmic scale
Note: Post-neutron isobaric element distributions in logarithmic scale. The experimental data
[59] (black symbols) are compared with the results of the GEF code (red symbols). The mass
numbers are specified in the figures.
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Figure 60: Isobaric Z distributions for
238
U(nth ,f), part 3, logarithmic scale
Note: Post-neutron isobaric element distributions in logarithmic scale. The experimental data
[59] (black symbols) and the evaluated data [1] (violet symbols) are compared with the results
of the GEF code (red symbols). The mass numbers are specified in the figures.
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Figure 61: Isobaric Z distributions for
238
U(nth ,f), part 4, logarithmic scale
Note: Post-neutron isobaric element distributions in logarithmic scale. The evaluated data
[1] (violet symbols) are compared with the results of the GEF code (red symbols). The mass
numbers are specified in the figures.
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Figure 62: Isobaric Z distributions for
238
U(nth ,f), part 5, logarithmic scale
Note Post-neutron isobaric element distributions in logarithmic scale. The evaluated data [1]
(violet symbols) are compared with the results of the GEF code (red symbols). The mass
numbers are specified in the figures.
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In Figures 63 to 67, the same data are shown in a linear scale. The range is adjusted
to the range of the data in each case individually.
Figure 63: Isobaric Z distributions for
238
U(nth ,f), part 1, linear scale
Note: Post-neutron isobaric element distributions in linear scale. The experimental data [59]
(black symbols) are compared with the results of the GEF code (red symbols). The mass
numbers are specified in the figures.
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Figure 64: Isobaric Z distributions for
238
U(nth ,f), part 2, linear scale
Note: Post-neutron isobaric element distributions in linear scale. The experimental data [59]
(black symbols) are compared with the results of the GEF code (red symbols). The mass
numbers are specified in the figures.
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Figure 65: Isobaric Z distributions for
238
U(nth ,f), part 3, linear scale
New: Post-neutron isobaric element distributions in linear scale. The experimental data [59]
(black symbols) and the evaluated data [1] (violet symbols) are compared with the results of the
GEF code (red symbols). The mass numbers are specified in the figures.
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Figure 66: Isobaric Z distributions for
238
U(nth ,f), part 4, linear scale
Note: Post-neutron isobaric element distributions in linear scale. The evaluated data [1] (violet
symbols) are compared with the results of the GEF code (red symbols). The mass numbers are
specified in the figures.
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Figure 67: Isobaric Z distributions for
238
U(nth ,f), part 5, linear scale
Note: Post-neutron isobaric element distributions in linear scale. The evaluated data [1] (violet
symbols) are compared with the results of the GEF code (red symbols). The mass numbers are
specified in the figures.
8.2.7
Energy dependence
In order to benchmark the GEF code up to 14 MeV, the fission yields of 3 masses are
compared with the available data. As suggested in Reference [153], the masses 111,
115 and 140 were chosen. In order to be comparable, the shell effect at symmetry was
set to +0.3 MeV for all systems, corresponding to a weak anti-shell. First, the ratio
Y (A = 115)/Y (A = 140) is shown in Figure 68. The GEF code is able to reproduce the
global trend for all systems. Also the absolute values agree very well for the systems 238 U
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Figure 68: Evolution of Y (A = 115)/Y (A = 140) with excitation energy
Note: Y (A = 115)/Y (A = 140) for different fissioning nuclei as a function of the excitation
energy. Experimental data [154, 155, 156, 157] are compared with the result of the GEF code.
and 239 Pu, while they are slightly overestimated for 235 U and 232 Th. Thus, the conditional
barrier for symmetric fission seems to be slightly higher for these two systems.
Two regions in energy domain should be studied: before the second-chance threshold,
where the fissioning nucleus is always the same but with different excitation energy and
after this threshold, where multi-chance fission must be taken into account. While Figure
68 shows the evolution of the valley/peak ratio, represented by the A = 115/A = 140
ratio, before the threshold (≈10-12 MeV) for neutron-induced fission of 232 Th, 235 U, 238 U,
and 239 Pu, the fission yields of two masses near symmetry (A = 111, A = 115) and one
mass near the heavy peak (A = 140) are compared in Figures 69 and 70 for the systems
235
U and 239 Pu in an extended energy range. Obviously, the values above the threshold
for second-chance fission (En ≈ 6 MeV) and third-chance fission En ≈ 12 MeV are well
reproduced. The calculation slightly underestimates the yields of the system 235 U near
symmetry above the threshold of second-chance fission, a deviation which is opposite to
the deviation found at lower energies revealed in Figure 68. However, the yield near the
asymmetric peak at A = 140 is underestimated in this low-energy range for both systems.
According to Tables 11 and 12, the calculated probabilities for first-chance fission at
E = 8 MeV and E = 14 MeV are somewhat higher, but still close to the values given in
the ENDF/B-VII library [141].
As the difference of some specific masses can be the result of some local effects, the
complete fission-yield distributions were studied for 4 MeV, 8 MeV and 14 MeV where
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Figure 69: Energy dependence of mass yields in
235
U(n,f)
Note: Measured fission yields of A = 111 (left), A = 115 (middle) and A = 140 (right) for
235 U(n,f) as a function of E [154, 158, 159] are compared with the GEF results. The hatched
n
band indicates the uncertainty of the calculated values.
Table 11: First-chance probability for
235
Energy
GEF first-chance
relative
probability
8 MeV
14 MeV
54.3 %
25.2 %
Table 12: First-chance probability for
239
U(n,f), En = 8 MeV and 14 MeV.
Library
first-chance
relative
probability
46.1 %
29.6 %
Pu(n,f), En = 8 MeV and 14 MeV.
Energy
GEF first-chance
relative
probability
8 MeV
14 MeV
80.4 %
59.5 %
Library
first-chance
relative
probability
65.6 %
44.8 %
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Figure 70: Energy dependence of mass yields in
239
Pu(n,f)
Note: Measured fission yields of A = 111 (left), A = 115 (middle) and A = 140 (right) for
239 Pu(n,f) as a function of E [155, 159, 160] are compared with the GEF results. The hatched
n
band indicates the uncertainty of the calculated values.
library evaluations are also available. Figure 71 shows the fission-yield distributions for
U at 4 and 8 MeV. The predicted fission-yield distribution at 4 MeV overestimates
the experimental data slightly. At 8 MeV, there exist two data-sets from different experiments. The GEF prediction is close to the symmetric data of Glendenin et al. but
higher than the data from Chapman et al.. This is in line with an analysis reported in
[153], where comparisons were made with the Ford experimental data which are consistent
with the Glendenin data, according to which the Chapman data seem to systematically
underestimate the symmetric part.
The fission yields of the system 239 Pu(n,f) at 4 and 8 MeV shown in Figure 72 are
rather well reproduced by the GEF code.
In addition to the comparison at the lower energies (4 and 8 MeV), the fission yields of
235
U and 239 Pu at En = 14 MeV are shown in Figure 73. Some deviations to the evaluated
data are found: The calculated fission yields at symmetry are slightly lower than the
evaluated values. However, in view of the scattering of the data from different experiments
this deviation is not very significant. Moreover, the calculated yield distribution shows a
more pronounced structure in the peak regions if compared to the evaluation. Also, this
deviation is not very significant due to the uncertainties and the large scattering of the
experimental data.
235
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Figure 71: Mass yields for
235
U(n,f), En = 4 MeV and 8 MeV
Note: Fission-yield distribution for 235 U(n,f) for En = 4 MeV and En = 8 MeV. Experimental
data [154, 161] are compared with the GEF results.
Figure 72: Mass yields for
239
Pu(n,f), En = 4 MeV and 8 MeV
Note: Fission-yield distribution for 239 Pu(n,f) for En = 4 MeV and En = 8 MeV. Experimental
data [155] are compared with the GEF results.
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Figure 73: Mass yields for
235
U(n,f) and
239
Pu(n,f), En = 14 MeV
Note: Fission-yields distributions for 235 U(n,f) and 239 Pu(n,f) at En =14 MeV. The result of the
GEF model is compared with values from data libraries and experimental data. Colour points
correspond to experimental data [158, 160, 159]. The calculated values are given together with
their estimated uncertainties
8.2.8
Discussion
8.2.8.1 Mass distributions: Figures 31 to 56 demonstrate an overall rather good
agreement between the empirical mass distributions and the results of the GEF code.
In particular, the variation of the global shape for different systems and as a function
of energy, which can be rather drastic in some cases, are rather well reproduced. When
looking in detail, however, more or less severe discrepancies can also be found for several
systems. In this comparison, it must be considered that the quality of the experimental
data that are shown in the figure or that were used for the evaluation may differ strongly
from one system to another one. In many cases, the evaluation is based on incomplete
data of limited quality due to the difficulties of the experiment. In other cases, there are
plenty of high-quality data. The mass distributions from double-energy or double-velocity
measurements generally suffer from a limited resolution. The data from spontaneous
fission of the heaviest nuclei are important for revealing the strong variation of the global
shape from system to system, but the uncertainties are rather large, e.g. due to low
statistics. Thus, a careful analysis is needed to decide whether the discrepancies between
empirical data and calculated spectra are to be attributed to shortcomings of the model
or to uncertainties of the empirical data.
A first step towards a quantitative assessment is the determination of the reduced
Chi-squared values of the differences between empirical data and calculated values. The
Chi-squared values were only determined for the evaluated data, because they are mostly
based on radio-chemical methods with full identification of the fission-product mass, while
the mass spectra from kinematical measurements are distorted by the limited resolution.
These Chi-squared values are listed in Table 8.
The Chi-squared values are also shown in an histogram in Figure 74. The distribution
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has a main peak around unity, containing 50 of the 59 cases. It reaches from 0.3 to 1.7
and, thus, seems to be essentially in agreement with the expected scattering caused by
the uncertainties of the evaluated data. This picture already does not give indications
for a shortcoming of the model in these 50 cases which represent 85 % of the cases.
The uncertainties of the model seem to contribute little to the Chi-squared values of the
systems in the main peak, because this peak centres at about unity without taking the
uncertainties of the model into account. The remaining 9 cases will be investigated in
more detail.
Figure 74: Chi-squared deviations for mass distributions
Note: Chi-squared deviations of the mass distributions from GEF calculations from the empirical
data (Figures 31 - 56) in a logarithmic binning. The height of the histogram represents the
number of cases per bin.
The largest Chi-squared value is observed for the system 227 Th(nth ,f). An inspection of
the figure, in particular in the logarithmic scale, reveals that the evaluated spectrum has
a very unusual shape that differs substantially from the other spectra of near-by systems:
The descent from the asymmetric mass peaks towards symmetry is exceptionally gradual.
This observation is a strong argument for assuming that the problem is caused by an
unrealistic result of the evaluation in this case.
The second-highest Chi-squared value is found for the system 229 Th(nth ,f). Also in
this case, the largest deviations occur on the inner wings of the asymmetric peaks. This
time, the slope agrees, but the borders towards symmetry are shifted in the calculation.
In this case, there exist very reliable and precise data from different sources, including an
experiment at the Lohengrin spectrograph [148]. Thus, this problem must be attributed
to a shortcoming of the model. This displacement of the inner wing of the asymmetric
mass peak with respect to the global description of the model, which agrees in practically
all other cases, is very astonishing. It indicates a local effect that is not considered in the
model.
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Seven other systems show larger Chi-squared values between 3.2 and 9.5: 238 U(sf),
248
Cm(sf), 253 Es(sf), 235 U(nth ,f), 251 Cf(nth ,f), 254 Es(nth ,f), and 255 Fm(nth ,f). The deviations for 238 U(sf) are not severe and look unsystematical. 248 Cm(sf), 253 Es(sf), 251 Cf(nth ,f),
254
Es(nth ,f), and 255 Fm(nth ,f) form a group of nuclei that seem to suffer from incomplete
data and/or large uncertainties. A closer look reveals two abnormalities: All systems
show rather schematic shapes at the outer wings of the mass distributions that differ
substantially from the spectrum of 252 Cf(sf) which has been investigated in great detail.
In addition, 254 Es(nth ,f) and even more clearly 255 Fm(nth ,f) show a shift of the minimum
around symmetry with respect to the calculation. The mass distribution of 255 Fm(nth ,f)
is symmetric around A=128 ± 0.5, which is half the mass of the fissioning system. Thus,
there is no room for neutron evaporation, although the systematics suggests a mean
prompt-neutron yield around 5. Finally, 235 U(nth ,f) is a very peculiar case. For this
nucleus, the measurements are so precise that the experimental uncertainties are appreciably smaller than the general uncertainties of the model calculation. Thus, although
the evaluated mass spectrum is very well reproduced by the calculation, relatively small
deviations lead to a large Chi-squared value.
In summary, from the 59 evaluated mass distributions, we found one case where a
shortcoming of the model is clearly proven. In 6 cases, it seems that the evaluation suffers
from poor data. In addition, the uncertainties of the evaluation have been underestimated,
causing large Chi-squared values. A closer look to these cases does not give indications
for a shortcoming of the model but rather for somewhat faulty evaluations. In one case,
the measured yields (and thus the evaluated data) are so precise that the uncertainties
of the model exceed the uncertainties of the evaluation substantially. This leads to large
Chi-squared values, although the mass distribution is well reproduced.
A closer view to the mass distributions reveals some additional somewhat minor problems either in the evaluation or in the model. The calculated yields around symmetry
often deviate from the empirical data. The prediction of the low yields at symmetry is
very demanding due to their high sensitivity to excitation energy and the strong variation
from system to system. This is particularly critical for fast-neutron-induced fission, where
the neutron-energy distribution in the experiment might be rather broad, and the energy
distribution of fissioning systems is weighted with the energy-dependent fission probability of the specific system. Moreover, experimental data in the region of very low yields
near symmetry are very scarce, and the uncertainties are large. In the right wing of the
left peak for the system 237 Np(nth ,f) appears a structure, which is probably caused by a
contamination of the target by a heavier nucleus. Note that the position of the heavy
fission-product group is roughly independent of the fissioning nucleus, while the position
of the light fission-fragment group moves accordingly. There are several systems, where
the outer wings of the evaluated mass distribution appear to have a schematic, unrealistic shape, probably due to the lack of reliable data (in addition to the systems already
mentioned above): 250 Cf(sf), 232 U(nth ,f), and 237 Np(n,f) at En = 14 MeV are the most
prominent cases.
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8.2.8.2 Charge polarisation and emission of prompt neutrons: There is a rather
limited amount of data on the neutron excess of the fission products. Figure 57 proves that
the mean neutron excess and the fluctuations are well reproduced over the large range from
233
U(nth ,f) to 249 Cf(nth ,f). The reason for the deviations for Apost > 105 from 249 Cf(nth ,f)
is not clear, because the resolution of the experiment was insufficient to distinguish the
energy-loss signals from the different elements. The data show very nicely the influence of
an even-odd staggering, predominantly in the Z distribution. The good reproduction of
the mass-dependent mean prompt-neutron multiplicity for 237 Np(n,f) as a representative
for a lighter system and 252 Cf(sf) as a representative for a heavier system that will be
discussed in Section 8.4 shows that the influence of charge polarisation and promptneutron emission is correctly modelled in the GEF code. In particular, the transport
of the additional energy from the 5.55 MeV neutron to the heavy fragment is correctly
reproduced [98].
8.2.8.3 Nuclide distributions: The isobaric Z distributions shown in Figures 58 to
67 demonstrate a very good agreement of the GEF calculations with the measured data
for the system 235 U(nth ,f). For almost all mass chains, the error bars of the evaluation
and the error bars from the estimated uncertainties of the model calculation overlap.
Due to the good agreement of the mean value and the standard deviation of the isobaric
Z distributions also for other systems shown in Figure 57 one expects that the nuclide
distributions of other systems are described with a similar quality.
8.2.8.4 Energy dependence: The relative intensities of the fission fragments at symmetry are most sensitive to the excitation energy of the fissioning system. The general
increase of the valley-to-peak ratio of the mass distributions is rather well described by
the GEF model. This validates the statistical approach assumed for the population of the
fission channels, including the parametrisation of the level densities. There are mostly
minor deviations in the absolute values, but they do not seem to be systematical.
On the basis of this analysis one can expect that the GEF model is suited to give
reliable estimations of the complete fission-fragment yields at higher energies, at least up
to excitation energies around 20 MeV, where experiments are scarce and in most cases
very incomplete.
8.2.9
Conclusion and outlook
The overall quality of the GEF code for predicting the fission-product yields was demonstrated on the basis of all mass distributions of the ENDF/B-VII evaluation and other
data, comprising measured fission-product mass distributions, mass-dependent promptneutron yields and mass-chain Z yields. Severe shortcomings of the model appeared only
for the system 229 Th(nth ,f), while deficiencies of the evaluation were found for a number
of other systems.
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From the careful comparison of the evaluated data and the predictions of the GEF
code it becomes evident that the GEF code can be applied to substantially improve the
quality of evaluated data.
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8.3
Isomeric yields
The angular-momentum distribution cannot be directly measured and is often extracted
from the isomeric ratio. In order to reduce the bias due to the model used for the
extraction of the angular momentum, the only benchmark on the prediction of the angular
momentum detailed here are the isomeric ratios.
The Naik compilation [162, 163] was used as a reference for experimental data. These
isomeric ratios are usually extracted from γ-ray spectroscopy coupled with radio-chemistry
technique. This technique relays on the values of the branching ratio Iγ which are often
known with an uncertainty larger than 5-10%, which consequently leads to large uncertainties on the isomeric ratio. Moreover, the nuclei studied have a long life time (> 1
minute) and are in the heavy peak. Very few measurements were performed on the light
peak. As the angular momentum depends on the mass of the fragment, new measurements
on the light peak will be welcome.
The isomeric ratio predicted by the GEF model depends on the mass of the fragment,
the deformation of the fragment, the Z parity of the fragment, the excitation energy, the
spin of the compound nucleus, and on the spin difference between the isomeric state and
the ground state. These dependencies will be studied in this section.
Figure 75 represents the ratio of the isomeric yield (Ym ) over the sum of the isomeric
Figure 75: Isomeric ratios for
Note: Measured isomeric ratios for
results.
239 Pu(n
th ,f)
239
Pu(nth ,f).
[162, 163, 164] are compared with the GEF
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yield and the ground-state yield (Ym + Ygs ) for the 239 Pu(nth ,f) reaction. The GEF prediction agrees with the experimental data within the 1σ- uncertainty in the majority of
cases. The agreement does not depend on the Z parity : the odd-Z isomeric ratios (Sb, I,
Cs, La) are predicted with the same quality as the even-Z isomeric ratios (Te, Xe). The
quality does not seem to be influenced by the mass of the fragment, at least on the heavy
peak. The spin difference, defined as Spin(isomer) − Spin(groundstate) is nearly always
the same (values around four in most cases), so the influence of this difference cannot be
studied. However, it can be seen on the chain of the Sb isotopes that a negative spin
difference is not problematic for the GEF model.
Figure 76: Isomeric ratios for odd-Z compound nuclei
Note: Measured isomeric ratios for odd-Z compound nuclei: 237 Np (5/2+) (n,f),
(n,f), 243 Am (5/2+) (n,f) [165] are compared with the GEF results.
241 Am
(5/2-)
Figure 77: Isomeric ratios for even-Z compound nuclei
Note: Measured isomeric ratios for even-Z compound nuclei:
(0+) (sf) [162] are compared with the GEF results.
232 Th
(0+) (n,f), 235 U (7/2-) (n,f),
252 Cf
The ratios of the high-spin yield (Yh ) over the sum of the low-spin and the highspin yields (Yl + Yh ) were compared for 6 different nuclei in Figures 76 and 77: 237 Np
(5/2+) (n,f), 241 Am (5/2-) (n,f), 243 Am (5/2+) (n,f), 232 Th (0+) (n,f), 235 U (7/2-) (n,f),
252
Cf (0+) (sf). The conclusions are the same as the previous ones on 239 Pu(nth ,f). The
agreement between the experimental data and the GEF prediction is good whatever the
parity and the spin of the compound nucleus.
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In order to extend our benchmark of the GEF prediction as a function of the compound
nucleus, four isomeric ratios (the Sb chain and 135 Xe) are compared for 15 fissioning
systems in Figures 78 and 79. The 132 Sb isomeric ratios are clearly over-predicted whereas
the 128 Sb ratios are under-predicted. The 135 Xe isomeric ratio is well reproduced. In each
case, the tendency with the variation of the compound nucleus is good.
Figure 78: Isomeric ratio for Sb isotopes.
Note: Measured isomeric ratios for the Sb chain for different fissioning nuclei [162, 163] are
compared with the GEF results.
Even if few data are available for the light group, Figure 80 shows that the GEF
code tends to overestimate the few ones available on the average. According to the
experimental data on the heavy peak that indicate a small influence of the compound
nucleus on the isomeric ratio, the experimental 99 Nb value for 235 U can be wrong as the
experimental 99 Nb value for 239 Pu is the complementary to 1. This can be due to the
inversion Ym /Ygs = Yl /Yh for the nucleus contrary to a lot of nuclei where Ym /Ygs = Yh /Yl .
In order to study the influence of the excitation of the compound nucleus on the
isomeric ratio, the 135 Xe and 133 Xe isomeric ratios are also compared as a function of the
excitation energy with the Ford measurement (thermal, 2 MeV, 14 MeV) in Figure 81.
Ford observed an increase of the Ym /Yg ratio for 133 Xe whereas he saw no increase for
135
Xe. The GEF code does not reproduce the nearly constant behaviour before 3 MeV.
A larger number of data is however required to extend this observation more especially
in the range En = 2 − 14 MeV.
Photo-fission reactions also give some hints that the excitation energy does not have
a huge influence on the isomeric ratio at least in the range E ∗ = 9.7 − 14.1 MeV (En =
4 − 8 MeV). Figure 82 shows the isomeric ratio for 134 I for 235 U and 238 U as a function
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Figure 79: Isomeric ratio for
Note: Measured isomeric ratio for
with the GEF result.
135 Xe
135
Xe
for different fissioning nuclei [162, 163] is compared
Figure 80: Isomeric ratios for light fragments
Note: Measured isomeric ratios for light fragments from different thermal-neutron-induced reactions [166, 164, 167] are compared with the GEF results.
of the excitation energy of the compound nucleus. The experimental ratios are nearly
constant for both compound nuclei in the domain E ∗ = 9.7 − 14.1 MeV. The GEF
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Figure 81: Energy dependence of isomeric ratios for
133
Xe and
Note: Measured isomeric ratios for 133 Xe and 135 Xe from 235 U(n,f) and
neutron energies [168] are compared with the GEF results.
135
Xe
239 Pu(n,f)
for different
predictions are also nearly constant. The excitation-energy dependence of the isomeric
ratio as parametrised in the GEF code seems to be correct. The absolute values, however,
are substantially underestimated.
Figure 82: Isomeric ratio for
134
I from photofission
Note: Measured isomeric ratios for 134 I from 235 U(γ,f) and
energies [169] are compared with the GEF results.
238 U(γ,f)
at different excitation
In conclusion, the GEF prediction is in very good agreement with the data in a large
number of cases.
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8.4
Prompt-neutron multiplicities
The multiplicities of prompt fission neutrons contain valuable information and, thus,
provide a stringent test for the understanding of the fission process. Moreover, this
quantity is very important for nuclear technology. The prompt-neutron multiplicity is
rather directly connected with the excitation energies of the fragments. Fortunately,
prompt-neutron yields have been measured for many fissioning systems. In a few cases,
the variation of the prompt-neutron yield as a function of excitation energy and fragment
mass has been determined.
8.4.1
System dependence
There exist extended systematics of prompt-neutron multiplicities for spontaneous fission
and for thermal-neutron-induced fission. They are compared in Figures 83 and 84 with
results of the GEF code. It is obvious that the data cannot be parametrised by a simple
function of a macroscopic parameter, e.g. the fissility parameter Z 2 /A or the Coulomb
parameter Z 2 /A1/3 .
8.4.1.1 Spontaneous fission: In spontaneous fission, the most striking structural effects are the horizontal slope for the Pu isotopes that deviates from the average slope of
the other isotopic chains and the decrease towards the heaviest Fm isotopes. The first
effect is the consequence of the large yield of the standard 1 fission channel, which is
characterised by a 17 MeV higher TKE value [143] and a correspondingly reduced TXE.
Let us remind that the large yield of the S1 fission channel for these nuclei is attributed
to the influence of a shell in the light fragment around Z = 42 in the GEF model. The
yield of the standard 1 channel increases gradually from 236 Pu to 244 Pu, which explains
the almost constant prompt-neutron multiplicity for the Pu isotopes. The reduction of
the prompt-neutron yield due to the increasing yield of the S1 channel compensates the
general trend that shows an increase of the prompt-neutron yield with increasing mass
number, as can be seen in the behaviour of the uranium, curium, and californium isotopic
sequences. The second effect reflects the rather sudden appearance above 256 Fm of a
narrow symmetric fission component with TKE values which are higher by about 30 MeV
[170].
The measured values are very well reproduced by the GEF model with a few exceptions. The experimental value for 232 Th has a large uncertainty, and the one for 253 Es
was reported without mentioning the uncertainty range. Thus, these values may be considered with some caution. Moreover, the increase of the measured values from 256 Fm
to 257 Fm seems to be in conflict with the increase of the measured yield of the narrow
symmetric component and its high total kinetic energy, because the TKE and the TXE
are connected through the Q value by energy conservation. The expected further decrease
of the prompt-neutron yield towards 258 Fm is demonstrated by the calculated value in
Figure 83. Therefore, the measured value for 256 Fm may be doubted. The rms deviation between the remaining 19 experimental values and the corresponding calculations
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Figure 83: Systematics of prompt-neutron multiplicities for spontaneous fission
Note: Measured mean prompt-neutron multiplicities for spontaneous fission (black full symbols)
as a function of the mass number of the fissioning nucleus [171] in comparison with the result
of the GEF model (red open symbols). Experimental error bars are not shown when they are
smaller than the symbols. The value for 253 Es is reported without an experimental uncertainty.
amounts to 0.086. This is also the order of magnitude of the expected uncertainty for the
predictions of the prompt-neutron yields of nuclei in the vicinity of the systems shown in
Figure 83. Thus, the GEF model is expected to be able to estimate the prompt-neutron
multiplicity for spontaneous fission with a precision better than 0.1 units.
8.4.1.2 Thermal-neutron-induced fission: In the case of thermal-neutron-induced
fission, the situation is more complex. A number of data are rather well reproduced
by the GEF model, see Figure 84, but there are also large deviations. The value for
232
U reported in ref. [171] deviates by exactly one unit from the value obtained by the
GEF model. Unfortunately, ref. [171] cites another publication [173] that is not easily
accessible. Therefore, the possibility of a misprint, which is tentatively assumed in Figure
84, could not be verified. For the large discrepancies for 229 Th, 233 U, 238 Pu, 241,243 Am,
and 245,247 Cm there is no obvious explanation. There is no obvious systematics in these
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Figure 84: Systematics of prompt-neutron multiplicities for nth -induced fission
Note: Measured mean prompt-neutron multiplicities for thermal-neutron-induced fission as a
function of the mass number of the target nucleus [172] (black full symbols), [171] (blue shaded
symbols), and [141] (green open symbols) in comparison with the result of the GEF model (red
open symbols). We assumed that the value 3.132 for 232 U given in [171] (blue open symbol) is
wrong due to a misprint. The tentatively corrected value (2.132) is marked by a blue shaded
symbol. Experimental error bars are not shown when they are smaller than the symbols.
deviations. It is striking that the data for the following systems with easily available
target material (235,238 U, 237 Np, 239,241 Pu, 252 Cf), and also 232 U are very well reproduced.
The situation is not clear. More experimental work would be desirable in order to
better understand the structural effects, which are eventually responsible for the observed
deviations, and in order to verify the result and to exclude possible systematic uncertainties of one or the other experiment. The rms deviation between all 21 experimental
values, including those with large error bars, amounts to 0.17, which is about twice the
value found for spontaneous fission. For 232 U, the tentatively corrected value was used.
Thus, the GEF model is expected to be able to estimate the prompt-neutron multiplicity
for thermal-neutron-induced fission with a precision better than 0.2 units.
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8.4.2
Energy dependence
For a few target nuclei, the prompt-neutron multiplicity has been measured in neutroninduced fission as a function of the incident-neutron energy. Great part of these data
are compared with the results of the GEF model in Figure 85. The experimental data
are taken from ref. [171]. Only part of the data are shown if they overlap in order not
to overload the figures. The overall slope of the neutron multiplicity as a function of
neutron energy is well reproduced by the model. The data for the two systems 235 U(n,f)
and 239 Pu(n,f), which have been studied most extensively, are very well reproduced over
the whole energy range up to almost 30 MeV. The data for 232 Th(n,f) show a structure
at the onset of second-chance fission, which is well reproduced by the model as well. The
strong increase of the neutron multiplicity just above the threshold for second-chance
fission can be explained by the fact that second-chance fission is only possible in this
energy range, if the kinetic energy of the emitted pre-fission neutron is so low that the
excitation energy of the daughter nucleus falls above its fission barrier. Thus, the average
prompt-neutron energy is exceptionally low, and the corresponding neutron multiplicity
is exceptionally high. Also, another peculiarity of this system, the weak increase of the
prompt-neutron multiplicity in the low-energy range up to 3 MeV is present in the model
results. This effect is connected with the fact that for this even-even nucleus low incident
neutron energies lead to excitation energies around the fission barrier. In the tunneling
regime, at energies below the fission barrier, the TXE values do not directly follow the
variations of the initial excitation energy. A similar, however much weaker structure than
in 232 Th(n,f) at the onset of first-chance fission of 238 U(n,f) in the calculated values is not
seen in the data. The structures seen in the model results at the threshold for third-chance
fission near 15 MeV for 235 U(n,f) and 239 Pu(n,f) cannot be compared, because there are
no data measured between 15 and 22 MeV.
It is interesting to note that the energy-dependent prompt-neutron multiplicity is
perfectly reproduced by the GEF model for the odd-A targets 235 U(n,f) and 239 Pu(n,f),
in contrast to the even-A targets 232 Th(n,f) and 238 U(n,f), where the neutron yield is
overestimated above the threshold for second-chance fission. This problem is probably
connected with the difficulties in describing the fission probabilities of systems with relatively low fissility and high neutron-separation energies, which were already reported in
Section 8.1. In general, the specific behaviour of the prompt-neutron multiplicity at the
onset of a higher-chance fission strongly depends on the behaviour of the fission probability around the fission threshold, which shows a gradual increase in part of the systems
and a more or less pronounced peak structure in other systems. In the first case, the mean
neutron energy tends to increase, in the second case it tends to decrease with the opening
of another fission chance. This feature strongly depends on structural effects in the level
density (see the discussion in Section 8.1). In addition, the neutron yield for 232 Th(n,f)
is underestimated at incident-neutron energies below 5 MeV. Possibly, this problem is related in some way with the discrepancies observed in the fragment yields from the fission
of several thorium isotopes.
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Figure 85: Energy dependence of mean prompt-neutron multiplicities
Note: Measured prompt-neutron multiplicity for 232 Th(n,f), 235 U(n,f), 238 U(n,f), and 239 Pu(n,f)
(black symbols, different symbols are used for different experiments) as a function of neutron
energy (data from ref. [171]) in comparison with the result of the GEF model (red line).
8.4.3
Fragment-mass dependence
In the actinides, the prompt-neutron multiplicity has the typical saw-tooth behaviour as a
function of fragment mass. Figure 86 shows the measured data for the system 237 Np(n,f)
for two incident-neutron energies. The data for 252 Cf(sf) are shown in Figure 87. The
data are rather well reproduced by the GEF model.
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Figure 86: Variation of the mass-dependent prompt neutron yield with E ∗
Measured prompt-neutron yield in 237 Np(n,f) as a function of pre-neutron mass at two different
incident-neutron energies [58] (data points) in comparison with the result of the GEF model
(histograms).
Figure 87: Mass dependent prompt-neutron yield in
252
Cf(sf)
Measured prompt-neutron yield in 252 Cf(sf) [174] as a function of pre-neutron mass (data points)
in comparison with the result of the GEF code (dashed line). The experimental uncertainties
are smaller than the symbols.
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There are two prominent features in the model: First, the increasing yields are caused
by the fragment deformation at scission which increases with the fragment mass in the
range of the light and in the range of the heavy fragments. This feature is a consequence
of a general characteristics of shells in deformed nuclei, already mentioned in Section 6.4:
these shells extend over a broad range of neutron, respectively proton number, but the
optimum deformation is correlated with the size of the system [53, 55]. Secondly, the
intrinsic excitation energy at scission is subject to energy sorting [98]. Thus, the higher
incident neutron energy in 237 Np(n,f) leads to an increased neutron yield in the heavy
fragment, only.
It is remarkable that the data of the two systems are well reproduced by the model
with the fundamental assumption that the fragment deformation at scission is a unique
function of the fragment shells.
8.4.4
Multiplicity distributions
The distribution of prompt-neutron multiplicities provides a test for the fluctuation of
the total excitation energy of the fragments. In the GEF model, the largest contribution
to these fluctuations is caused by the distribution of fragment deformations around the
equilibrium value at scission. The distributions for 239 Pu(nth ,f) and 252 Cf(sf) shown in
Figure 88 are perfectly reproduced, whereas the calculated distribution for 235 U(nth ,f) is
slightly too narrow.
Figure 88: Distribution of prompt-neutron multiplicities
Note: Measured distribution of prompt-neutron multiplicities in 235 U(nth ,f), 239 Pu(nth ,f) and
[175, 176] (black full points) in comparison with the result of the GEF model (red open
points).
252 Cf(sf)
8.4.5
Conclusion
The manifold data on prompt-neutron multiplicities show a large variety of gross and
subtle features. The GEF model is able to reproduce most of them with a satisfactory
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quality. Even more importantly, the model traces these features back to peculiar aspects
of the physics governing the fission process. This way, the model provides a link to other
observables which are consistently described by the model.
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8.5
8.5.1
Prompt-neutron energies
Key systems
8.5.1.1 Prompt-neutron spectra The experimental prompt-fission-neutron spectra
for the systems 235 U(nth ,f) [177] and 252 Cf(sf) [178] are compared with results of the GEF
code in Figure 89. In order to better visualise the deviations, the lower panels show a
reduced presentation with the spectra normalised to a Maxwellian distribution with the
parameter T = 1.32 MeV.
In this calculation, the de-excitation of the separated fragments has been obtained
within the statistical model. It is assumed that both the emission of neutrons and the
emission of E1 gammas do not change the angular momentum on the average, which seems
to be a good approximation in the relevant angular-momentum range [70]. When the
Figure 89: Prompt-neutron spectra for
235
U(nth ,f) and
252
Cf(sf)
Note: Experimental prompt-fission-neutron spectra (black lines and error bars) for 235 U(nth ,f)
[177] (left panels) and 252 Cf(sf) [178] (right panels) in comparison with the result of the GEF
model (red lines) in logarithmic scale. In the lowest panels, all spectra have been normalised to
a Maxwellian with T = 1.32 MeV.
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yrast line is reached, the angular momentum is carried away by a cascade of E2 gammas.
The inverse neutron absorption cross-section has been described by the parametrisation
from [115]. Since the fast-neutron spectrum in fission is composed of the contributions
from many emitting fragments, the use of this global description that is computed very
quickly is probably not too critical. Gamma competition at energies above the neutron
separation energy was considered. The gamma strength of the giant dipole resonance
(GDR) following the description proposed in ref. [116] was applied. The nuclear level
density was modelled by the constant-temperature description of v. Egidy and Bucurescu
[44] at low energies. The level density was smoothly joined at higher energies with the
modified Fermi-gas description of Ignatyuk et al. [45, 46] for the nuclear-state density:
ω∝
√
√
π
ãU )
exp(2
12ã1/4 U 5/4
(73)
with U = E +Econd +δU (1−exp(−γE)), γ = 0.55 and the asymptotic level-density √
pa2/3
rameter ã = 0.078A+0.115A . The shift parameter Econd = 2 MeV −n∆0 , ∆0 = 12/ A
with n = 0, 1, 2, for odd-odd, odd-A and even-even nuclei, respectively, as proposed in
ref. [43]. δU is the ground-state shell correction. A constant spin-cutoff parameter
was used. The matching energy is determined from the matching condition (continuous
level-density values and derivatives of the constant-temperature and the Fermi-gas part).
Values slightly below 10 MeV are obtained. The matching condition also determines a
scaling factor for the Fermi-gas part. It is related with the collective enhancement of the
level density (see Section 3.2).
The resulting prompt-neutron spectra are shown in Figure 89. The transformation of
the neutron-energies into the laboratory frame was performed considering the acceleration
phase [179, 180] after scission by a numerical trajectory calculation. The mean pre-scission
total kinetic energy was assumed to be 40% of the potential-energy gain from saddle to
scission derived by Asghar and Hasse [86] as:
< T KE >pre = 0.032(Z 2 /A1/3 − 1527)M eV
(74)
with a standard deviation of the same amount. The distribution was truncated at negative
values.
The good reproduction of the measured neutron spectra, especially for the lighter
system 235 U(nth ,f), does not give indication for additional neutron emission at scission
[181, 182, 183, 184].
The emission during the acceleration phase is stronger for the system 252 Cf(sf), since
higher excitation energies and, thus, shorter emission times are involved in this system.
Neutron emission during fragment acceleration reduces especially the laboratory energies
of the first neutrons emitted at short times from the most highly excited fragments in
252
Cf(sf) and allows for a decently consistent description of the two systems with the
GEF code, using the same parameter set. Experimental prompt-fission-neutron spectra
of the systems 239 Pu(nth ,f) and 240 Pu(sf) are compared with the result of the GEF code
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in Figures 90 and 91, again using the same model parameters. Obviously, the data are
very well reproduced.
Figure 90: Prompt-neutron spectrum for
239
Pu(nth ,f)
Note: Experimental prompt-fission-neutron spectrum for the system 239 Pu(nth ,f) from ref. [185]
(black open symbols) and from [186] (blue full symbols) in comparison with the result of the
GEF code (red thick full line). The calculated spectrum was normalised to the measured total
neutron multiplicity (ν̄ = 2.88 [175]). The measured spectra are slightly scaled for minimising
the overall deviations from the calculated spectrum in order to better compare the spectral
shapes.
In general, the GEF code reproduces the available experimental fission-prompt-neutron
spectra rather well. This qualifies the GEF code for estimating prompt-neutron spectra in
cases where experimental data do not exist. These data can be generated by downloading
the code [54] and by performing the calculations for the appropriate fissioning system.
The code also seems to be a suitable tool for improving evaluations.
8.5.1.2 Correlations: Since the prompt-neutron spectra measured in the laboratory
frame are the result of a convolution due to the emission under different angles from
the moving fragments, they are not very sensitive to the yield of neutrons with very low
energies in the frame of the fragments. Therefore, one may look for other experimental
signatures that are more sensitive to specific features of the neutron emission. One of
these signatures is the variation of the neutron multiplicity as a function of the angle
between the directions of the emitted neutrons and the light fission fragment. Figure 92
shows the experimental data [188] in comparison with the result of the GEF code. The
measured data are well reproduced over almost the complete angular range. The code
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Figure 91: Prompt-fission-neutron spectrum for
240
Pu(sf)
Note: Experimental prompt-fission-neutron spectrum for the system 240 Pu(sf) from ref. [187]
(black symbols) in comparison with the result of the GEF code (red line). The measured data
were scaled to the height of the calculated spectrum. Since the experiment covers especially well
the lower-energy range, a double-logarithmic presentation was chosen.
underestimates the yield only very close to the direction of the light fragments. The two
right-most points of the distribution correspond to angles of 5.7 and 9.9 degrees, corresponding to neutron energies in the fragment frame of 30 and 10 keV, respectively. Thus,
these deviations can be explained by a slight underestimation of the neutron-absorption
cross-sections in the very restricted low-energy regime below 50 keV. In the promptneutron spectrum, Figures 89 to 91, these events appear at laboratory energies around 1
MeV due to the velocity of the emitting fragment. Here, no indication for this deviation
can be seen. It seems that the description by the GEF code is very well suited for estimating the prompt-neutron spectra in the laboratory frame of heavy fissioning systems,
which are most important for technical applications. The slight deviations in the angular
distributions, Figure 92, have practically no influence on the energy distribution of the
prompt neutrons in the laboratory frame.
In the following, we investigate the prompt-neutron yield as a function of the fissionfragment total kinetic energy. Figure 93 shows a comparison of the result of the GEF code
with experimental data [189, 190] and a previous calculation of Kornilov [191]. The GEF
calculation has been performed using Thomas-Fermi masses of Myers and Swiatecki [20]
with recommended shell corrections and schematic even-odd fluctuations. The variation
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Figure 92: Prompt-neutron multiplicities versus neutron direction
Note: Variation of the prompt-neutron multiplicities versus the neutron direction relative to the
light fission fragment. The result of the GEF code is compared with experimental data from
ref. [188]. The nominal threshold in the experiment was 0.15 to 0.2 MeV.
of the prompt-neutron yields from the light and the heavy fragment are assumed to be
uncorrelated for a given split in Z and N .
The GEF calculation, in particular the slope, is rather close to the experimental data
in the region between 155 MeV and 185 MeV. Also, the low-energy point of Boldeman
et al. is well reproduced. For energies higher than 185 MeV, all calculations, also the
calculation of Kornilov, are appreciably below the experimental data. The cut-off of the
neutron multiplicity slightly below 200 MeV is probably realistic, because even for the
splits with the highest Q values the excitation energies of the fragments fall below the
corresponding neutron separation energy for these high TKE values.
One should not forget that scattering phenomena can considerably disturb experimental data in regions of low yield as e.g. demonstrated in ref. [192]. Such processes would
tend to flatten the variation of the measured prompt-neutron yield as a function of TKE.
In this context, it is interesting to note that the data of Boldeman et al. have a steeper
slope than the data of Vorobyev et al., especially in the wings of the TKE distribution.
The data of Vorobyev et al. even extend to TKE values, where there is hardly any yield
expected, and neutrons are still seen above TKE = 200 MeV, where neutron emission is
suppressed in the GEF code due to the Q-value limit. This puts also doubts on the data
of Vorobyev et al. for total kinetic energies below 150 MeV, where the yield is low, and
scattering phenomena may have an important influence.
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Figure 93: Mean prompt-neutron yield as a function of TKE
Note: Mean prompt-neutron yield as a function of fission-fragment total kinetic energy for the
system 235 U(nth,f). The experimental data of Boldeman et al. [189] and Vorobyev et al. [190]
are compared with a calculation of Kornilov [191] (labelled as LD Ignatyuk) and the result of the
GEF model (red line). The lower part shows a zoom on the central part of the TKE distribution.
The green histogram shows the calculated pre-neutron TKE distribution in an arbitrary scale.
The dotted vertical lines denote the region that contains 95 % of the fission events.
The GEF code reproduces also well the measured mean prompt-neutron yields as a
function of the total fission-fragment total kinetic energy for spontaneous fission of 252 Cf
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Figure 94: Mean prompt-neutron yield as a function of TKE for
252
Cf(sf)
Note: Mean prompt-neutron yield as a function of fission-fragment total kinetic energy for the
system 252 Cf(sf). The experimental data of Budtz-Jorgensen et al. [174] are compared with
the result of the GEF model. The green histogram shows the calculated pre-neutron TKE
distribution in an arbitrary scale.
of ref. [174], see Figure 94. The deviations at high TKE appear in a region of extremely
low yield. They may be explained by a background of events with lower TKE due to
random coincidences of fragment and prompt-neutron signals in the experiment. Also
the deviations at low TKE appear in a region with low yield. They may be caused at
least to a part by incompletely measured TKE values due to scattering phenomena in the
experiment.
One may speculate that the transport of a multitude of correlations along the fission
process in the GEF code without any intermediate averaging has an important influence
on correlations between different fission observables. These correlations might not have
been fully considered in other models. The calculations with the GEF code do not give
strong hints for additional phenomena like scission neutrons; the data of Figures 93 and
94 can rather well be reproduced with the assumption of prompt-neutron emission from
the fragments after scission, only.
8.5.1.3 Conclusion: The GEF model reproduces a large variety of observables with a
good precision in a consistent way without further adjustment to specific fissioning systems
with a unique parameter set. With this global approach one is able to predict several
characteristic quantities of the fission process, e.g. the energy and multiplicity distribution
of prompt-fission neutrons, without the need for specific experimental information of the
respective system, e.g. measured mass-TKE distributions. All properties of the fission
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fragments that are considered in the code (e.g. nuclear charge, mass, excitation energy,
angular momentum) are sampled in the corresponding multi-dimensional parameter space
by a Monte-Carlo technique. Thus, all respect ive correlations are preserved. Moreover,
correlations between all observables considered in the code are provided on an event-byevent basis. It should be stressed that it is straightforward to deduce covariances for the
calculated prompt-neutron spectrum determined by the inner logic of the GEF model in
analogy to the covariances of the fission-fragment yields from the list-mode data of the
perturbed-parameter calculations.
The measured prompt-neutron spectra in fission induced by thermal neutrons are very
well reproduced by the GEF code without any specific adjustment of the model for all
systems that were investigated. It is to be expected that this agreement is preserved for
fission induced by neutrons of higher energies. There are no systematic deviations which
suggest the presence of scission neutrons in these cases.
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8.5.2
Energy dependence
8.5.2.1 Introduction: This section deals with the description of prompt-neutron spectra in neutron-induced fission reactions over a larger excitation-energy range extending
from spontaneous fission to multi-chance fission. A number of measured prompt-neutron
spectra from elaborate experiments are compared with the results of the GEF code
[131, 54]. The GEF code calculates the contributions from the excited nucleus before scission and from the fragments simultaneously with the statistical model in a consistent way
together with many other fission observables. The calculation is done without using an analytical formula with adjustable parameters for the shape of the prompt-neutron spectrum
and without any input on fission-fragment properties for specific systems. Therefore, this
study aims to give a coherent picture on the variation of the prompt-neutron spectrum
for different fissioning systems as a function of excitation energy.
8.5.2.2 Description of the calculation: The following figures show comparisons of
measured fission prompt-neutron spectra extracted from EXFOR with results of the GEF
code [131, 54]. All measurements have been performed relative to the system 252 Cf(sf).
Thus, the data marked as ratio or R are directly measured. If the deduced prompt-neutron
spectra are also given in EXFOR, they are shown as well, marked as yield or Y. The scale
is dN /dE in units of 1/MeV.
GEF calculations on neutron yields and energy distributions have been performed for
the indicated systems and for 252 Cf(sf). All calculations have been performed without
any adjustment to specific systems with the very same parameter set. No particular information from experimental data, e.g. A-TKE spectra, has been used. The GEF model
exploits three general laws of dynamics, quantum mechanics and statistical mechanics in
order to model the fission process in a comprehensive and consistent way with a modest input of empirical information and a minimum of computational effort: the influence
of inertia and friction on the fission dynamics is implicitly considered by a dynamical
freeze-out, the influence of nuclear structure is traced back to the early influence of fragment shells, and the transport of thermal energy between the fragments before scission is
assumed to be driven by entropy.
In order to clearly distinguish the calculation of prompt-neutron yields with the general
approach of the GEF model from other models, a short summary of alternative approaches
seems to be appropriate. One of the first widespread descriptions of the prompt-neutron
spectrum was introduced by Watt [193]. He proposed a closed formula, deduced from a
Maxwell-type energy spectrum from one or two average fragments and the transformation
into the frame of the fissioning system with at least two adjustable parameters: the
temperature and the velocity of the average fragment. The ”Los-Alamos model” [194]
extended this approach essentially by the use of a triangular temperature distribution
of the fragments to a four-term closed expression for an average light and an average
heavy fragment. A similar two-fragment model was also used by Kornilov et al. in [181].
In 1989, Madland et al. [195] introduced the point-by-point model by considering the
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emission from all individual fragments, specified by Z and A. This model was further
developed e.g. by Lemaire et al. [196], Tudora et al. [197] and Vogt et al. [198]. In refs.
[199, 200, 201, 202], the spectral shape was parameterised by the Watt formula [193] or an
empirical shape function that had been introduced by Mannhart [203] in order to better
model the shape of the neutron energy spectra in the fragment frame. Kornilov [204]
proposed a phenomenological approach for the parameterisation of a model-independent
shape of the prompt-neutron spectrum. This approach was later also used by Kodeli et
al. [205] and Maslov et al. [206]. These models often reach a high degree of agreement
with the measured prompt-neutron spectra for particular fissioning systems with especially
adjusted parameters. All models cited above are based on empirical data: the Watt model
and the Lost-Alamos model are directly fitted to the measured prompt-neutron spectrum,
while the point-by-point model is based on the measured A-TKE distribution. Manea et
al. [207] proposed a scission-point model that predicts the TKE(A) distribution, in order
to allow for calculations of prompt-neutron spectra with the boint-by-point method if only
the mass distribution is known. For completeness, we also mention a paper of Howerton
[208], who developed a method for predicting (Z, A, En ) distributions. The required input
values are the charge and mass numbers (Z and A) and the binding energy of the last
neutron in the (A + 1) nucleus. This method was used in [209].
As a result of the GEF model, the prompt-neutron spectra and the ratios to the
calculated 252 Cf(sf) spectrum are shown. Note that the measured and calculated spectrum
for 252 Cf(sf) are shown in Figure 89. Due to the Monte-Carlo method used in the GEF
code, the spectra show statistical fluctuations, especially in the high-energy tail. The
calculated total prompt-neutron multiplicity is given in addition in the figures. Note that
the deviations between GEF results and experimental data in the two representations
(ratio and yield) are not consistent, because the GEF yield ratios and the experimental
yields (measured yield ratios times neutron yields for 252 Cf(sf)) have been obtained with
different prompt-neutron reference spectra: For the GEF ratios the calculated 252 Cf(sf)
spectrum was used, for the experimental yields an evaluated 252 Cf(sf) spectrum was used.
It seems that most of the experiments aimed only to determine the shape of the spectra.
Therefore, an arbitrary scaling factor was applied, such that the total prompt-neutron
multiplicity agrees approximately with the GEF result. These scaling factors are listed in
the legends of the figures. All figures are shown in logarithmic and in linear scale.
8.5.2.3 Results:
Spectra:
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Figure 95: Prompt-neutron spectrum from
232
Th(n,f), En = 2.9 MeV
Note: Data from EXFOR dataset 411100081.
Figure 96: Prompt-neutron spectrum from
232
Th(n,f), En = 14.7 MeV
Note: Data from EXFOR dataset 411100081.
Figure 97: Prompt-neutron spectrum from
233
U(nth ,f)
Note: Data from EXFOR dataset 40871013.
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Figure 98: Prompt-neutron spectrum from
235
U(n,f), En = 100 K
Note: Data from EXFOR dataset 31692006.
Figure 99: Prompt-neutron spectrum from
235
U(nth ,f)
Note: Data from EXFOR datasets 40871011, 40871012.
Figure 100: Prompt-neutron spectrum from
238
U(n,f), En = 2.9 MeV
Note: Data from EXFOR dataset 411100101.
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Figure 101: Prompt-neutron spectrum from
238
U(n,f), En = 5 MeV
238
U(n,f), En = 6 MeV
238
U(n,f), En = 7 MeV
Note: Data from EXFOR dataset 41450003.
Figure 102: Prompt-neutron spectrum from
Note: Data from EXFOR dataset 41447003.
Figure 103: Prompt-neutron spectrum from
Note: Data from EXFOR dataset 41447003.
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Figure 104: Prompt-neutron spectrum from
238
U(n,f), En = 10 MeV
Note: No data available.
Figure 105: Prompt-neutron spectrum from
238
U(n,f), En = 13.2 MeV
238
U(n,f), En = 14.7 MeV
Note: Data from EXFOR dataset 41450003.
Figure 106: Prompt-neutron spectrum from
Data from EXFOR dataset 411100101.
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Figure 107: Prompt-neutron spectrum from
239
Pu(nth ,f)
Note: Data from EXFOR datasets 40871009, 40871010, 40872006, 41502004.
Figure 108: Prompt-neutron spectrum from
240
Pu(sf)
242
Pu(sf)
Note: Data from EXFOR dataset 414210021.
Figure 109: Prompt-neutron spectrum from
Note: Data from EXFOR dataset 414210031.
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Figure 110: Prompt-neutron spectrum from
241
Am(n,f), En = 2.9 MeV
241
Am(n,f), En = 4.5 MeV
Note: Data from EXFOR dataset 415890021.
Figure 111: Prompt-neutron spectrum from
Note: Data from EXFOR dataset 415890031.
Figure 112: Prompt-neutron spectrum from
241
Am(n,f), En = 14.6 MeV
Note: Data from EXFOR dataset 415890041.
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232 Th(n,f ),
En =2.9 MeV: The measured spectrum is very well reproduced up to 7
MeV. At higher energies, the measured spectrum shows strange fluctuations, which points
at experimental uncertainties.
232 Th(n,f ), E =14.7 MeV: Most part of the spectrum is very well reproduced by
n
the calculation. However, there is a local enhancement at very low energies, which is not
strong enough in the calculation below 0.5 MeV. The structure around 8 MeV is narrower
and slightly shifted in the calculation.
233 U(n ,f ): The calculated spectrum is very well reproduced in the range between
th
0.8 and 4.7 MeV that is covered by the experiment. (The spectrum was not normalised.)
235 U(n,f ), E =100 K: The spectrum is well reproduced over the whole energy range.
n
Between 1 MeV and 5 MeV, the calculated spectrum is a little bit lower. (This spectrum
is not normalised.)
235 U(n ,f ): The measured spectrum is very well reproduced over the whole energy
th
range.
238
U(n,f ), En =2.9 MeV: The measured spectrum is very well reproduced in the
energy range below 6 MeV. At higher energies, the measured spectrum shows strange
fluctuations, which points at experimental uncertainties.
238 U(n,f ), E =5 MeV: The measured spectrum is very well reproduced up to 5 MeV.
n
At 5 MeV there is a kink in the measured data, and the data have a smaller slope at higher
energies. This may point at a background in the experiment.
238 U(n,f ), E =6 MeV: The calculated spectrum above 1 MeV has a steeper slope than
n
the measured one. In addition, the calculated spectrum is enhanced at the lowest energies
due to a contribution from second-chance fission. This enhancement is overestimated by
the calculation.
238 U(n,f ), E =7 MeV: The calculated spectrum above 1 MeV has a slightly steeper
n
slope than the measured one. The spectrum is enhanced at the lowest energies due to a
contribution from second-chance fission. Amplitude, width and position of this structure
are not correctly reproduced by the calculation.
238 U(n,f ), E =10 MeV: This spectrum, for which no data are available, is added in
n
order to allow a systematic view on the variation of the structure caused by the threshold
of second- chance fission.
238 U(n,f ), E =13.2 MeV: The spectrum is well reproduced by the model within the
n
experimental uncertainties. However, there is a local enhancement at very low energies,
which is not strong enough in the calculation. The structure due to the threshold of
second-chance fission is slightly shifted to lower energies and narrower in the calculation.
238 U(n,f ), E =14.7 MeV: Again, there is a local enhancement at very low energies
n
below 0.6 MeV, which is not strong enough in the calculation. The shape and the position
of the structure due to the threshold of second-chance fission are not correctly reproduced
by the calculation.
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Figure 113: Prompt-neutron spectrum from
242
Am(nth ,f)
Note: Data from EXFOR dataset 414210081.
Figure 114: Prompt-neutron spectrum from
243
Am(n,f), En = 2.9 MeV
243
Am(n,f), En = 4.5 MeV
Note: Data from EXFOR dataset 415890051.
Figure 115: Prompt-neutron spectrum from
Note: Data from EXFOR dataset 415890061.
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Figure 116: Prompt-neutron spectrum from
243
Am(n,f), En = 14.6 MeV
Note: Data from EXFOR dataset 415890071.
Figure 117: Prompt-neutron spectrum from
243
Cm(nth ,f)
Note: Data from EXFOR dataset 415890081.
Figure 118: Prompt-neutron spectrum from
244
Cm(sf)
Note: Data from EXFOR dataset 413400041.
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Figure 119: Prompt-neutron spectrum from
245
Cm(nth ,f)
Note: Data from EXFOR dataset 414210091.
Figure 120: Prompt-neutron spectrum from
246
Cm(sf)
248
Cm(sf)
Note: Data from EXFOR dataset 413400051.
Figure 121: Prompt-neutron spectrum from
Note: Data from EXFOR dataset 41113004.
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239 Pu(n
There are two experimental results with different slopes of the highenergy tail. The slope of the calculated spectrum agrees better with the steeper slope
of one of the experiments, although this spectrum shows strong local fluctuations. The
steeper slope is also much closer to the ones of the systems 238 U(n,f), En =2.9 MeV and
246
Cm(sf), which have similar total prompt-neutron yields as 239 Pu(nth,f). Since all these
cases are restricted to first-chance fission, one should expect that the total prompt-neutron
yield is a measure of the mean excitation energies of the primary fragments, which means
that it should be correlated with the slope of the high-energy tail of the prompt-neutron
spectrum. Due to this argument, the spectrum with the steeper slope appears to be more
likely the correct one.
240 Pu(sf ), 242 Pu(sf ): The measured spectra are well reproduced by the calculation,
if the fluctuations in the experiment at higher energies are disregarded.
241 Am(n,f ), E =2.9 MeV: Below 4 MeV, the measured spectrum is well reproduced
n
by the calculation. A comparison at higher energies is difficult due to the strong fluctuations in the measured spectrum.
241 Am(n,f ), E =4.5 MeV: The measured spectrum is well reproduced by the calcun
lation below 2.5 MeV. Above this energy the experimental data fluctuate rather strongly.
241 Am(n,f ), E =14.6 MeV: The measured spectrum is very well reproduced by the
n
calculation, including the structure around 9 MeV.
242 Am(n ,f ): Below 4.5 MeV, the measured spectrum is very well reproduced by the
th
calculation. A comparison is difficult at higher energies due to the strong fluctuations in
the measured spectrum.
243 Am(n,f ), E =2.9 MeV: The measured spectrum is well reproduced below 4 MeV.
n
The calculated spectrum is much softer in the high-energy tail than the measured one.
It is remarkable that the measured spectrum is appreciably stiffer than the spectrum of
252
Cf(sf), although the total prompt neutron yield is almost the same. This points at an
experimental problem.
243 Am(n,f ), E =4.5 MeV: The measured spectrum is well reproduced below 4 MeV.
n
A comparison at higher energies is difficult due to the strong fluctuations of the measured
spectrum.
243 Am(n,f ), E =14.6 MeV: The measured spectrum is generally well reproduced by
n
the calculation. The structure around 8 MeV is slightly shifted to lower energies.
243 Cm(n ,f ): When comparing the measured and the calculated ratios to the specth
trum of 252 Cf(sf), the calculated spectrum appears to be much softer than the measured
one. It is astonishing that the measured spectrum is as stiff as the one for 243 Am(n,f) at
En =14.6 MeV which has a much higher total prompt-neutron yield. The measured spectrum is also much stiffer than the one of the system 252 Cf(sf), although the total promptneutron yield is about the same. However, when comparing the empirical prompt-neutron
spectrum, already multiplied with the reference spectrum of 252 Cf(sf), which is also listed
in EXFOR, with the calculated spectrum, in particular in logarithmic scale in the right
panel, there is very good agreement. That means that the ratio to 252 Cf and the spectrum
given in EXFOR are not consistent. This kind of inconsistency is not observed for any
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other case.
244 Cm(sf ):
The measured spectrum is well reproduced below 6 MeV. At higher
energies, the measured spectrum has an unusual shape with a dip around 9 MeV. This
dip is not found in the calculated spectrum.
245 Cm(n ,f ): The measured spectrum is well reproduced at energies below 2.5 MeV.
th
One value at 3 MeV seems to be in error. At higher energies, the measured spectrum
shows strong fluctuations, making a comparison difficult.
246 Cm(sf ) and 248 Cm(sf ):
Both measured spectra are well reproduced by the calculation up to 4 MeV. There are deviations and fluctuations in the experiment at higher
energies.
8.5.2.4 Discussion:
General observations: The most salient features of this comparison are:
1. There is a qualitatively rather good reproduction of the shape of the spectra, including the structural effects. There are some deviations in the quantitative reproduction
of the structure at the threshold of second-chance fission.
2. In some cases, the exponential slope of the calculated spectrum exceeds the slope
of the measured spectrum. The most important deviations are found for 238 U(n,f),
En =5 and 6 MeV, 239 Pu(nth ,f) with respect to one experiment, 243 Am(n,f), En =2.9
MeV, and 243 Cm(nth ,f).
3. There are some fluctuations in the data for which the model does not provide an
explanation. The most severe cases are 241 Am(n,f), En =2.9 MeV, 242 Am(nth ,f),
243
Am, En =4.5 MeV, and 243 Cm(nth ,f).
4. Two experiments for
239
Pu(nth ,f) give diverging results.
5. There are some inconsistencies in different data tables from the same experiment
for 243 Cm(nth ,f) . There is very good agreement of the calculated prompt-neutron
spectrum with the spectrum, while there are strong deviations to the ratio with
respect to 252 Cf(sf).
Pre-fission neutron emission: The pre-fission neutrons are registered in coincidence with fission only if the excitation energy of the residual nucleus falls above its
fission barrier. This causes a pronounced structure in the prompt-fission-neutron spectrum. The structure of the calculated spectrum reproduces the structure in the measured
spectra rather well in most cases. In the calculations, the structure depends on the description of pre-scission neutron emission, pre-equilibrium and statistical, as well as on
the excitation-energy-dependent fission probabilities of the different nuclei. In particular, the mean energy of the structure in the calculated spectra depends on the value of
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the fission threshold in the GEF code. In particular for even-even fissioning nuclei, the
number and the nature of levels at the fission barrier below the pairing gap are subject
to strong nuclear-structure effects [126] and difficult to model with a global approach. In
the experiment, the width of this structure is very sensitive to the energy spread of the
incoming neutrons and the energy resolution in the measurement of the emitted neutrons.
The mean energy is very sensitive to the energy definition of the incoming neutrons.
Inverse cross section: Since the evaporation spectrum is calculated with a modified
Weisskopf formalism where the angular momentum is explicitly considered, the massand energy-dependent transmission coefficients for neutron emission were parametrised
by using inverse capture cross-sections according to Dostrowsky et al. [210] in a slightly
modified version for fast computing, as already mentioned in Section 3.10.7
Since the fast-neutron spectrum in fission is composed of the contributions from many
emitting fragments, the use of this global description is probably a satisfactory approximation.
8.5.2.5 Conclusion: The model behind the GEF code is unique in the sense that
it provides practically all observables from nuclear fission without any needs for specific
experimental information by using a single fully consistent model description for all heavy
fissioning systems. The present comparison with measured prompt-neutron spectra shows
good agreement in most cases, but also some deviations, mostly in the high-energy tail
of the spectrum and in the structures caused by threshold effects in pre-fission neutron
emission. These structures are not exactly reproduced by the calculation, although their
integral strength and their position in energy deviate only little in most cases. In particular
in the fission of the lighter systems at higher energies, the model does not provide enough
intensity at very low energies, mostly below 0.5 MeV, in the frame of the fissioning system.
Some of this additional intensity is explained by the emission during the acceleration
phase, but this contribution does not reach far enough down in energy. There seems to be
a source of very low-energetic neutrons with an exponential-like spectrum in the frame of
the fissioning system, which is not accounted for in the model. This problem has already
been discussed in [209, 199]. A possible origin of these low-energy neutrons could be the
pre-acceleration emission from fragments with very large transmission coefficients at low
energies, which are not accounted for in the global description used in the present model.
A systematic view on the experimental data suggests that the uncertainties are underestimated in several cases. There are strange fluctuations in the measured spectra for 241 Am(n,f), En =2.9 MeV, for 242 Am(nth ,f), for 243 Am(n,f), En =4.5MeV, and
for 245 Cm(nth ,f). Contradictory results were obtained from different experiments for
239
Pu(nth ,f). In the energy range up to En = 7 MeV, where at least most part of the
spectrum is only fed by first-chance fission, the high-energy tail of the measured spectra
7
The present version of the GEF code is conceived as a very fast code. Whenever possible, fast
algorithms were used as long as their approximations do not exceed the estimated general uncertainties
of the model. They may easily be replaced by more elaborate descriptions in the freely accessible code.
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becomes in general stiffer with increasing energy of the impinging neutron. This trend is
weaker in the calculated spectra in some cases. But the variation of the stiffness is not
continuous in the data as a function of the incoming-neutron energy. Sometimes, e.g. for
238
U(n,f) at En = 7 MeV, the spectrum becomes softer again with increasing energy of
the incoming neutrons. Moreover, the variations from one system to another one are not
consistent with the model. After a careful analysis of this problem, the situation appears
to be unclear. On the one hand, the mean temperature of the emitting fragments is
expected to increase with increasing incoming-neutron energy. Thus, the trend to stiffer
prompt-neutron spectra found in the experiment is qualitatively expected. On the other
hand, these experiments are certainly very challenging, and some results may suffer from
an incompletely suppressed background of scattered neutrons. This might be the reason
for some unexpected fluctuations of the logarithmic slope of the spectra from one system
to another as a function of incoming-neutron energy or total prompt-neutron yield. More
data of high quality would certainly be helpful for a better understanding of this problem.
We would like to point out a slight inconsistency between the good agreement of the
ratios of most prompt-neutron spectra to the spectrum of 252 Cf(sf) in this section and
the deviations found in the previous section between the measured and the calculated
spectrum of 252 Cf at energies below 5 MeV, while the spectrum of 235 U(nth ,f) was well
reproduced. This finding suggests to perform a deeper analysis of the experimental results.
One may conclude that the GEF model provides a global view on the systematic variation of the fission observables as a function of the fissioning system and its excitation
energy. It reproduces the measured prompt-neutron spectra in general rather well. A
detailed analysis reveals three types of deviations that are found for some of the systems:
the description of the structure in the prompt-neutron spectrum due to the contribution
of second-chance fission suffers probably from difficulties in modelling the level densities
of even-even nuclei below the pairing gap by the global approach used in the code. Furthermore, there seems to be a source for the emission of neutrons with very low energies
in some systems before or slightly after scission that is not sufficiently accounted for in
the model. Finally, we think that there are indications that the stiffness of the measured
prompt-neutron spectra is distorted in several cases by an incompletely suppressed background of scattered neutrons. Predictions for other systems where no experimental data
are available are expected to be possible with rather good quality.
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8.6
Prompt-gamma emission
In Figure 122, the calculated prompt-gamma spectrum for the system 235 U(nth ,f) is compared with the experimental data of [211]. One can distinguish the signatures of the
different contributions. The statistical E1 emission dominates the high-energy part above
2 MeV. E2 emission from rotational bands at the yrast line strongly fills up the spectrum
below 2 MeV. The structure in the calculated gamma spectrum in this energy range that
is caused by the energies of the rotational and vibrational transitions is not fully visible
due to the 100-keV binning of the spectrum. The amount of E2 emission is constrained
by the angular-momentum distribution of the fission fragments. The deviation of the
calculated from the measured spectrum at very low energy is probably explained by efficiency losses of the gamma detection to a great part. Internal conversion does not seem
to play a major role, because the gamma spectrum for 252 Cf(sf) shown in Figure 123 is
well reproduced in this low-energy range.
Figure 122: Prompt-gamma spectrum for
235
U(nth ,f)
Note: Experimental prompt-gamma spectrum for 235 U(nth ,f) [211] (black line) in comparison
with the result of the GEF model (red line). The calculated contribution from E2 radiation is
shown separately (blue dashed line).
Detailed experiments with very high counting statistics and large-volume, high-granularity
detectors, e.g. with the Darmstadt-Heidelberg Crystal ball, have been performed for spontaneous fission of 252 Cf. These experiments cover a γ-energy range up to 80 MeV including
the whole GDR and extending to the postulated radiation from nucleus-nucleus coherent bremsstrahlung of the accelerating fission fragments [212], which is not considered in
the GEF model. Figure 123 shows an overview on these data in comparison with the
result of the GEF code up to 15 MeV. Obviously, the features of this spectrum are fairly
well reproduced, in particular the contribution of E2 gammas below 3 MeV and the kink
near 8 MeV, approaching the peak energy of the GDR. The measured spectra have not
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been unfolded for the detector response. This explains most of the discrepancies between
measured and calculated spectra above 9 MeV [215]. In the lower-energy range (below 6
MeV), the data from [216, 217, 218, 219] are very close to the other experimental data
shown.
Figure 123: Prompt-gamma spectrum for
252
Cf(sf)
Note: Experimental prompt-gamma spectrum for 252 Cf(sf) (data points and black lines) in
comparison with the result of the GEF code (thin red line). Black dashed line: Raw spectrum
from [213], gate on the mass of the heavy fragment 126 ≤ AH ≤ 136. Black full line: Raw
spectrum from [213], gate on 144 ≤ AH ≤ 154. Full symbols: Raw data from [214]. The
calculated contribution from E2 radiation is shown separately (dashed blue line).
Several theoretical studies of the many complex features of these data have been
performed, mostly with modified versions of the CASCADE code [220], see e.g. refs. [221,
222]. A recent effort to simulate the spectrum of prompt fission gammas [223] exploited
detailed empirical knowledge on the spectroscopic properties of the fission fragments.
These calculations are complementary to the calculations with the GEF model: the models
mentioned are based on measured A-TKE distributions and prompt-neutron multiplicities
for determining the yields and the excitation energies of the fission fragments after scission
as a starting point of their calculation. In contrast, the GEF model treats the whole
fission process in a consistent and global way and does not use any particular empirical
information for a specific system.
Comparing the total gamma energies and even more the gamma multiplicities is delicate, because they strongly depend on the lower threshold of the gamma detection [224]
and eventually the branching of internal conversion, which is not considered in the GEF
model.
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8.7
Fragment kinetic energies
Another main fission observable is the kinetic energy of the fragments. The global shape of
the kinetic energy as a function of fragment mass is easily reproducible: the kinetic energy
can be estimated by the Coulomb repulsion between the deformed fragments. Describing
the kinetic energy with high precision is however difficult.
The kinetic energies of pre- and post-neutron fragments are usually measured by the
2v technique and the 2E-technique, respectively. The 2E-technique is very often used
to extract the pre-neutron energies, however, a correction on mass-dependent promptneutron yields ν(A) must be applied. This ν(A) correction is often based on the Wahl
evaluation e.g. [225]. The kinetic-energy data are usually of great quality, and the precision on the mean values is supposed to be around 0.5 MeV. The energy resolution of
the detectors, however, is limited, leading to a mass resolution between 2 and 3 units
[225]. The kinetic energy is also measured at the Lohengrin spectrograph, where only
post-neutron fragments are available.
Figure 124: Mass-dependent mean kinetic energy for
233
U(nth ,f)
Note: The measured mean kinetic energy before evaporation (left) and after evaporation (right)
of prompt neutrons as a function of the mass of the fragment for 233 U(nth ,f) [225, 226] is
compared with the result of the GEF model.
Table 13: Mean TKE before prompt-neutron emission for well known systems
Nucleus
Recommended
value
GEF
233 U(n
th ,f)
170.1 ± 0.5
172.32
235 U(n
th ,f)
170.5 ± 0.5
172.04
239 Pu(n
th ,f)
177.9 ± 0.5
178.85
252 Cf(sf)
184.1 ± 1.3
188.14
Note: Mean TKE in MeV before evaporation of prompt neutrons for well known systems. The
recommended values are extracted from ref. [230], page 321.
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Figure 125: Mass-dependent mean kinetic energy for
235
U(nth ,f)
Note: The measured mean kinetic energy before evaporation of prompt neutrons as a function
of the mass of the fragment for 235 U(nth ,f) [225, 227, 228] is compared with the result of the
GEF model.
Figure 126: Mass-dependent mean kinetic energy for
239
Pu(nth ,f)
Note: The measured mean kinetic energy before evaporation of prompt neutrons as a function
of the mass of the fragment for 239 Pu(nth ,f) [229, 230, 231] is compared with the result of the
GEF model.
Figures 124 to 127 show the mean kinetic energy of the fragments for different fissioning
nuclei. The agreement between the GEF predictions and the experimental data is very
good in the thermal-neutron-induced fission of 233 U, 235 U and 239 Pu (see Figures 124-126).
However, the total kinetic energies of 252 Cf(sf) from the GEF model are higher by 4 MeV
than the measured data (see Figure 127 and Table 13). A wider analysis of this deviation
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can be found in Section 10.3.
Important deviations are also seen in the kinetic energy of neutron-induced fission of
232
Th (see Figure 128) especially in the A = 120-130 region i.e. in the border region of
the SL and the S1 fission channel. This problem will be further investigated in Section
10.2.
Figure 127: Mass-dependent TKE before prompt-neutron emission for
252
Cf(sf)
Note: The measured mean total kinetic energy before evaporation of prompt neutrons as a
function of the mass of the fragment for 252 Cf(sf) [192, 174] is compared with the result of the
GEF model.
Figure 128: Mass-dependent pre-neutron TKE for
232
Th(n,f)
Note: The measured mean total kinetic energy before evaporation as a function of the mass of
the fragment for 232 Th(n,f) [233] is compared with the result of the GEF model.
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Small deviations for 232 Th(n,f) can also be observed in the regions of the S1 and the
SL modes; they can be due to a wrong correction of prompt-neutron multiplicity (ν(A) for
the experimental data when extracted from 2E technique or to an underestimation of the
TKE(S1) in the modelling of the S1 mode. The fission of the 240 Pu compound nucleus,
either in neutron-induced fission or in spontaneous fission, gives some answers about the
energy contribution of the S1 mode as its yield contribution is different (see Table 14).
It was observed that nearly all total-kinetic-energy distributions in neutron-induced
fission have a shape close to a Gaussian with some skewness, which is well reproduced
by the GEF code, see Section 3.7. An example is shown in Figure 129 for the 239 Pu+n
reaction. For the spontaneous fission of 238 Pu, 240 Pu and 242 Pu a second component
appears due to the large weight of the S1 mode. The component is also seen in the GEF
calculated distribution as shown in Figure 129. The calculated distributions are slightly
shifted and systematically narrower than the measured ones. It has already been noted
in [232] that the data of Milton et al. should be increased by 4 MeV to correspond to the
recommended value for 239 Pu obtained in several measurements with the 2v technique.
Figure 129: Pre-neutron TKE distribution for
240
Pu(nth ,f) and
240
Note: Total-kinetic-energy distribution before prompt-neutron evaporation for
240 Pu(sf) [236, 230].
Pu(sf)
240 Pu(n
th ,f)
and
Though the mean TKE value is, in a lot of cases, in very good agreement, the dispersion of this value should also be studied. Figure 130 illustrates the variance of the
TKE distribution for neutron-induced reactions. The order of magnitude and the Z dependence is well reproduced by the GEF model, however, the variance predicted by the
GEF code is always lower than the experimental value. Because of energy conservation,
a too small width in TKE also implies a too small width in the neutron multiplicity. The
neutron-multiplicity distribution, as detailed in Section 8.4, is also slightly too narrow for
235
U(nth ,f), but the width agrees perfectly for 239 Pu(nth ,f) and 252 Cf(sf). In view of the
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Table 14: Weights of fission modes for
240
239
Pu(sf)
Pu + n
S1
16,2 %
7.7 %
239
Pu(nth ,f) and
S2
83.2 %
88.9 %
Note: Relative contributions of the different modes for
the GEF model.
240
Pu(sf)
SA
0.6 %
2.8 %
239 Pu(n
th ,f)
and
240 Pu(sf)
according to
good agreement of the measured prompt-neutron multiplicity distributions with the GEF
results, it may not be excluded that the influence of the experimental energy resolution
is underestimated to some extent when this effect is substracted from the width of the
measured TKE distribution.
Figure 130: Variance of the TKE distribution for neutron-induced fission
Note: The measured variance of the TKE distribution before prompt-neutron evaporation for
neutron-induced reactions [232] is compared with the result of the GEF model.
Moreover, the kinetic-energy distribution for each mass is narrower than the measured
ones as shown in Figure 131 for 252 Cf(sf).
Figure 132 shows the width of the kinetic energy for 233 U(nth ,f) before and after evaporation of prompt neutrons. In agreement with the previous conclusions, the calculated
width before evaporation is always narrower than the measured ones. It has to be noted
that the experimental data (e.g Martin and Faust ones on Figure 132 ) for the width of
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Figure 131: Mass-dependent width of the TKE distribution for
252
Cf(sf)
Note: The measured width of the total-kinetic-energy distribution for 252 Cf(sf) [234] is compared
with the result of the GEF model.
the kinetic energy distribution can show some discrepancies mainly due to the correction
of the influence of the target thickness on the width of the energy distribution. The situation is not so clear after evaporation, where the GEF results agree with the data of
Faust et al. for the light fragments. The calculated values are, however, smaller than all
measured values in the heavy group.
According to a lot of experiments (ref. [232], page 366) it was observed that when
increasing the excitation energy of the system the mean TKE does not change a lot.
For example for 235 U(n,f) TKE(5 MeV) - TKE(th) was observed to be around -1 MeV
±0.5 MeV [237, 238, 100] which represents 0.5% of the TKE(th). This difference is of 2
MeV for the 239 Pu(n,f) reaction. However, the influence of the excitation energy of the
system on the mean TKE is clearly overestimated by the GEF code as shown in Figure
133. Due to energy conservation, the number of neutrons emitted should also evolve too
much with the excitation energy of the system, by about 0.2 neutrons on 5 MeV. But
this is not so much seen in Figure 85 in Section 8.4. Possible explanations could be an
increased mean kinetic energy of the prompt neutrons or an enhanced gamma emission.
This demonstrates, how the interconnection between different fission observables can be
studied by the GEF model.
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Figure 132: Width of the TKE distribution for
233
U(nth ,f)
Note: The measured width of the kinetic-energy distribution for 233 U(nth ,f) before evaporation
(full symbols) and after evaporation (open symbols) [225, 235, 226] are compared with the result
of the GEF model.
Figure 133: TKE as a function of the incident-neutron energy for
235
U(n,f)
Note: Difference between the total kinetic energy (TKE(En )) and the thermal value (TKE(th))
as a function of the neutron energy for 235 U(n,f). Measured data [238] are compared with the
result of the GEF model.
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9
Data for application
Nuclear industry strongly relies on the values of some specific fission yields. A very short
overview of some important features of the nuclear-reactor industry, where fission yields
are important, is presented below.
9.1
Decay heat
The isotopic fission yields are used in order to evaluate the decay heat. In a lot of cases,
the decay data (Iγ , ...) are the main problem of the decay-heat prediction, however, as
shown in [239], fission yields are also of importance for the prediction.
The decay-heat calculation was performed for 235 U for a fission pulse at thermal energy,
see Figure 134. The GEF results as shown in the figure agree quite well with the JEFF ones
proving the quality of the GEF prediction. The discrepancies between the experimental
data [241] and the calculated decay heat are mainly due to the decay data [242].
Figure 134: Total decay heat for
Note: Total decay heat for a fission pulse for
[240] with different fission yields.
9.2
235 U(n
th ,f).
235
U(nth ,f)
The calculations were performed with
Delayed neutrons
In order to calculate the delayed fission-neutron yield, the code implemented in [243] and
the associated delayed-neutron-precursor values were used. This procedure was validated
for 235 U. When the JEFF 3.1.1 fission yields are used, the calculated value of 100 · νd is
1.61; the recommended value is 1.62.
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As the number of the main delayed neutron precursors is limited (87 Br, 137 I, 88 Br, 138 I,
93
Rb, 89 Br, 94 Rb, 139 I, 85 As, 98m Y, 93 Kr, 144 Cs, 140 I, 91 Br, 95 Rb, 96 Rb, 97 Rb), the delayedneutron yield allows observing some local discrepancies of the fission yields, which were
found to be in good agreement with the empirical data in Section 8.2 at the first order.
Figure 135: Influence of the odd-even effect on the delayed-neutron yield
Note: Influence of the odd-even effect on the 235 U(nth ,f) calculated delayed-neutron yield in
comparison with the recommended value [244].
Table 15: Delayed-neutron yields
235
NEA
GEF
(GEF-NEA)/NEA
U
thermal
1.62
1.72
+6.2 %
235
U
fast
1.63
1.64
+0.6 %
238
U
fast
4.65
4.40
-5.4 %
239
Pu
thermal
0.65
0.71
+9.2 %
239
Pu
fast
0.651
0.673
+3.4 %
Note: Delayed-neutron yields for well-known system. The NEA recommended values are extracted from [244]. GEF calculations were performed at En = 2 MeV for the fast values.
Moreover, the delayed-neutron precursors are usually odd-Z nuclei, the delayed neutrons are consequently a non-direct way to observe the even-odd effect. Figure 135 illustrates the influence of the odd-even effect on the delayed-neutron yields for 235 U. When
the odd-even effect is modified by multiplying the local odd-even effect obtained by the
GEF code by a scale factor larger (smaller) than one, the odd yields decrease (increase)
and then, consequently, the delayed-neutron yield decreases (increases), as observed in
Figure 135.
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The delayed fission-yield values were also compared with the recommended values for
well-known fissioning systems in Table 15. The GEF code over-estimates the delayedfission yield in each case.
The group (as defined by Keepin et al. in ref. [245]) repartition obtained with the
GEF fission yield is also compared with the GODIVA and IPPE measurement at 1 MeV.
This repartition is in good agreement with the measured ones.
Figure 136: Delayed-neutron yield for
235
U(n,f), En = 1 MeV
Note: Relative delayed-neutron yield for 235 U(n,f) at En = 1 MeV as a function of the group
number as defined by Keepin et al. [245], Annex 1.
Figure 137: Delayed-neutron yields for
237
Np(n,f),
Note: Delayed-neutron yields for 237 Np(n,f), 235 U(n,f) and
therein, in comparison with the GEF results.
235
U(n,f) and
238 U(n,f),
238
U(n,f)
from [244] and references
The energy dependence of the delayed-neutron yield was also studied for 237 Np, 235 U
and 238 U, see Figure 137. The experimental data show a constant behaviour up to 4 MeV
and then a sharp decreasing slope. However, the GEF results show a decreasing slope
whatever the energy domain. The slope is also lower than in the experimental results.
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Figure 138: Calculated mass yields for
235
U(n,f), En = thermal and 5 MeV
Note: Mass-yield distribution for 235 U(n,f) at thermal energy and at En = 5 MeV calculated with
the GEF code. The full points correspond to masses with a main delayed-neutron precursor.
Figure 139: Delayed neutron yield up to En = 14 MeV
Note: Delayed-neutron yield for different fissioning systems at fast energy and at 14 MeV. Data
from [244] are compared with the GEF results.
This slope should not only be associated with the odd-even effect because, when
increasing the excitation energy this effect is reduced so the delayed-fission-neutron yields
should increase, which is the opposite of the experimental observation. The slope is
essentially due to the decrease of the peak-to-valley ratio as a function of the excitation
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energy. Figure 138 shows the fission yields of the masses with at least one main delayedneutron precursor. The fission yields of these masses decrease with the excitation energy.
The constant behaviour is due to the competition between the decreasing odd-even effect
and the decreasing peak/valley [246].
There is very few data above En ≈ 6 MeV, so the energy dependence when multichance fission is involved is difficult to benchmark. However, some data are available at
14 ± 1 MeV. Figure 139 shows that the experimental data are sometimes inconsistent
among each other giving an increase or a decrease of the delayed neutron yields with the
excitation energy. When considering that the delayed-neutron emission probability (Pn )
does not change with excitation energy and that νd = Σ(Yi · P ni ), the fission yields clearly
confirm a decreasing tendency. The GEF results are consistent with the experimental
data.
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10
Validation and evaluation of nuclear data
The GEF model does not directly make quantitative predictions. It rather provides a
rigid theoretical framework that defines a stringent link among a few key properties of
the fissioning systems and practically all kind of fission observables. The quantitative
predictions of the model depend on the values of a limited number of parameters, which
are determined in a comprehensive agree rather well with great part of the large body of
various experimental data.
Considering the large number of several hundred systems that is covered by the model
(not considering the intricate variation of the fission observables with excitation energy)
and the enormous complexity of the fission observables already for one system, the number of about 50 adjustable model parameters (that means far below one parameter per
system) is remarkably small. These numbers elucidate that the model establishes strong
relations between the different observables of one system and between the same observables of different systems. Thus, the model possesses the following fundamental virtues
and constraints:
1. The model allows predicting the behaviour of a specific system without any particular experimental information.
2. The model cannot be adjusted to a peculiar feature of a specific system.
The adjustment of the parameter values is difficult mainly by two reasons. First, the
amount and diversity of measured fission observables is so huge that a complete survey is
practically impossible. Second, erroneous experimental results should be recognised and
excluded from the fit procedure. This is not an easy task. However, the good agreement
of the model results with the majority of the data considered in this report proves the
excellence of the basic concept of the GEF model and gives confidence in the reliability of
the results. Thus, we propose to go a step further by using the GEF model for validating
the experimental and evaluated data by verifying their consistency. In this way, the
GEF model is employed for improving the quality of nuclear data. Moreover, due to its
predictive power, the GEF model is used for extending the amount of nuclear data. The
feasibility of this ambitious aim will be demonstrated by a few examples.
10.1
Indications for a target contaminant
The first case to be investigated is the fission-fragment mass distribution of the system
237
Np(nth ,f). Figure 140 demonstrates that the evaluated spectrum can rather well be
explained by a 60 % (!) target contaminant of 239 Pu. An additional strong argument for
the presence of a heavier target contaminant is the mean value of the mass distribution
< A > = 118.03, which would let room for the prompt emission of 1.94 neutrons, only.
This is in contradiction with the measured value of ν̄ = 2.5218 [172].
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Figure 140: Evidence for a
239
Pu contaminant in a
237
Np target
Note: The fission-fragment mass distribution of the system 237 Np(nth ,f) from ENDF/B-VII
(black crosses with error bars) in comparison with the result of the GEF code for a pure 237 Np
target (upper figure, blue full points) and for a composite target (40 % 237 Np and 60 % 239 Pu)
(lower figure, blue full points). In addition, the contribution from the 239 Pu contaminant is
shown separately in the lower figure (open red symbols).
Thus, we found two convincing arguments for the presence of an important target
impurity in the measurement underlying the evaluation of the mass distribution of the
system 237 Np(nth ,f).
Similar considerations can be employed to investigate and eventually revise the mass
distributions of other systems, for example 254 Es(nth ,f) and 255 Fm(nth ,f), which showed
severe deviations from the GEF results (see Section 8.2).
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10.2
An inconsistent mass identification
The next problem to be investigated is the discrepancy in the kinetic energy of neutroninduced fission of 232 Th between the GEF model and the experimental data found in
Section 8.7. Figure 141 reveals that there is most probably a problem in the experimental
data. According to our understanding, the kink in the kinetic-energy curve is caused
by the transition from the SL to the lumped S1 and S2 fission-channel component with
increasing mass asymmetry. This transition can also be observed by a kink in the mass
distribution. In Figure 141 this transition occurs at A = 126 consistently in the mass
distribution and in the kinetic energies from the GEF model. Also, the evaluated mass
distribution for fast-neutron-induced fission is found at almost the same place. However,
the kink in the measured kinetic energies is shifted by about 5 units to lower masses. This
Figure 141: Inconsistency of mass and TKE for
232
Th(n,f), En = 2.9 MeV
Note: Fission-fragment mass distribution (upper part) and kinetic energies (lower part) of the
system 232 Th(n,f), En = 2.9 MeV from the GEF model (red symbols) together with the evaluated
mass distribution from ENDF/B-VII from fast-neutron induced fission of 232 Th (upper part,
black crosses) and with the measured kinetic energies [233] (lower part, blue triangles). The
dashed line marks the border of the asymmetric fission component according to the GEF model.
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finding evokes severe doubts on the reliability of this measurement. The same problem
appears for the measurement at En = 4.81 MeV of the same authors.
10.3
A problem of energy conservation in
252
Cf(sf )
Now we will have a closer look on the discrepancy found in the total kinetic energy for
spontaneous fission of 252 Cf documented in Section 8.7. Since this is one of the most
carefully studied system, the shift of 4 MeV found in the GEF result relative to the
experimental data must be considered very seriously.
However, there is a connection of the TKE with the prompt-neutron yield, the promptneutron energy spectrum, the prompt-gamma spectrum and the fission-fragment nuclide
distribution: the nuclide distribution defines the weights of the different fission Q values,
and the data on prompt-gamma and prompt-neutron emission define the total excitation energy of the fragments. Since all these observables are well reproduced by the
GEF model, as much as experimental data are available, the observed shift of 4 MeV is
surprising.
Since the prompt-neutron yield is most strongly correlated with the TKE, the empirical
values from different sources are compared with the GEF result in Table 16. There is good
agreement.
Table 16: Prompt-neutron multiplicity of the system
Model
ν̄
GEF
3.75
ref. [172]
3.759 ± 0.0048
ref. [171]
3.755
252
Cf(sf)
ref. [141]
3.88
Note: Mean prompt-neutron multiplicity of the system 252 Cf(sf) from the GEF code in comparison with the values from different evaluations.
The span between the different values from the different evaluations correspond to a
span of about 1 MeV in TXE. Assuming that the ENDF value [141] is correct, this would
explain 1/4 of the shift. Thus, there remains a real problem to describe all the interconnected observables in a consistent way, in particular in view of the good agreement of
the GEF results with the TKE values of other systems.
10.4
Complex properties of fission channels
It is remarkable that the GEF model is able to describe the fission-fragment distributions
and their kinetic energies for all fissioning systems with a unique set of four fission channels. This is in contrast to previous work, where a complex set of channels had to be used
in order to reproduce the experimental data. This becomes most evident for spontaneous
fission of 252 Cf. Table 17 compares different parametrisation on the basis of the Brosa
model [56] with the result of the GEF model.
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Table 17: Yields of fission channels for the system
Model
SL
S1
S2
SA
S3
SX
GEF
2.6E-3 %
0.54 %
85.93 %
13.53 %
—
—
Brosa [247]
(3.0 ± 0.2) %
(13.5 ± 0.5) %
(48.2 ± 1.1) %
(0.3 ± 0.1) %
(35.0 ± 1.2) %
—
252
Cf(sf)
Brosa [248]
3.1566 %
12.6676 %
46.9569 %
—
0.9284 %
36.2905 %
Note: Relative yields of the fission channels for the system
parametrisation.
252 Cf(sf)
according to different
252
Cf(sf) is one of the most intensively investigated systems. The mass distribution
that has been determined with high precision is reproduced by the GEF model with
a reduced Chi-squared of 0.62 with practically only 3 fission channels, the S1, S2 and
the super-asymmetric (SA) channel. The super-long channel is so weak that it does not
play any role. Also, the kinetic energies are well reproduced, except a general shift, see
dedicated sections.
The reason for the smaller number and the strongly different yields of the fission
channels obtained with the GEF model lies in the properties of the fission channels themselves. In the Brosa model [56], the shape of the mass distribution of all fission channels
is assumed to be Gaussian. The mean total kinetic energy is parametrised as
Y (T KE) = (
−(L − lmax )2
200 2
) · h · exp(
)
T KE
(L − lmin )ldec
(75)
The charge-asymmetry degree of freedom enters via:
L=
e20 Z(ZCN − Z)
T KE
(76)
As described in Section 6.2, in the GEF model the shape of the mass distribution of
the S2 fission channel is given by a rectangular distribution convoluted with Gaussian
distributions with different diffusenesses at the lower and the upper border. Moreover,
the variation of the TKE as a function of fragment mass does not only consider the
Coulomb factor Z1 · Z2 . The TKE is also influenced by the fact that the deformation of
the fragments in the different fission channels is mass dependent. This implies a different
behaviour of the mean TKE as a function of fragment mass.
These more complex properties of the fission channels explain the strongly different
relative yields of the fission channels in the GEF model and allows describing all systems
consistently with the same set of fission channels.
188
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10.5
Data completion and evaluation
In many cases, the experimental data are incomplete, and it is the task of an evaluation
process not only to estimate the reliability and consistency of the measured data but also
to estimate the missing values with the help of systematics or theoretical models. The
GEF model in combination with the dedicated optimisation code MATCH [10] offers an
efficient tool for this purpose.
The GEF code provides a complete set of fission-fragment yields for a specific fissioning
system with uncertainties and covariances between all individual yields as determined by
the model. Also many other observables with their uncertainties and covariances can be
obtained, see Section 7. If there are no experimental data, the GEF result may directly be
used as a set of estimated values e.g. in order to extend evaluated data tables. In many
cases, however, there are some experimental results available, but they are incomplete or
rather uncertain. In this case, the result of the GEF code can be used for complementing
the partial experimental data set in a consistent way. For this purpose, the GEF results
should be adjusted in a suitable way to the experimental data.
10.5.1
Mathematical procedure in two dimensions
In order to illustrate the solution of the problem, a schematic case in two dimensions
is presented. The result of the GEF code, fission-fragment yields with their uncertainties and the covariance matrix, defines a multi-variant normal distribution. This is a
multi-dimensional probability-density distribution of a Gaussian-shaped cloud. In two
dimensions, this distribution is given by:
f (x, y) =
+
1
q
2πσx σy 1 − ρ2xy
· exp(−
1
(x − µx )2
[
2(1 − ρ2xy )
σx2
(77)
(y − µy )2 2ρxy (x − µx )(y − µy )
−
])
σy2
σx σy
The variables are defined as follows:
– x and y form a two-dimensional space of possible values of two fission-fragment yields.
– µx and µy are the most probable values of the yields given by the GEF code.
– σx and σy are the standard deviations of the uncertainty ranges of these two yields given
by GEF.
– ρxy is the correlation coefficient between the variables x and y given by the GEF code.
– ρxy · σx · σy is the covariance between the variables x and y.
From this distribution, one can derive a Log-Likelihood function L that expresses the
likelihood of a set of fission-fragment yields x and y to be compatible with the GEF result:
L(x, y) =
1
(x − µx )2 (y − µy )2 2ρxy (x − µx )(y − µy )
[
+
−
]
2(1 − ρ2xy )
σx2
σy2
σx σy
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General description of fission, GEF model, (78)
189
Validation and evaluation of nuclear data
Let us assume that there is one experimental value xm available with the standard
deviation of the uncertainty range σm . The Log-Likelihood function Lm that expresses
the likelihood of a fission-fragment yield x to be compatible with the experiment is given
by:
Lm (x) = −
(x − xm )2
2
2σxm
(79)
A set of variables x and y that is best compatible with both the model and the experiment may be found by constructing a combined Log-Likelihood function Lc , essentially as
the sum of L and Lm and by searching the parameter values xc and yc that maximise the
combined Log-Likelihood function Lc . This way, the information of the model calculation
is considered in two ways: First, the absolute values deduced from the general knowledge
on the physics derived from the body of available data and, secondly, the covariances
that link the different yields by the inner logic of the model. However, this procedure
would give more weight to the model result. Therefore, the number of yields provided by
the model, in this case 2, and the number of measured yields, in this case 1, should be
considered by enhancing Lm accordingly.8
Thus, the proposed combined Log-Likelihood function with the appropriate weight is:
Lc = L + 2 · Lm
(80)
The exponential of the combined Log-Likelihood function (properly normalized) defines the resulting multi-variant normal distribution, that is the multi-dimensional probabilitydensity distribution of the fission yields, by which the corresponding covariance matrix is
defined
It is expected that the correlations inside the model already assure that the resulting
yields respect to a high degree some trivial conditions, e.g. that complementary yields
are equal or the sum over the yields is normalised. In the code, the correlations of the
model can be enhanced by increasing the last term in the bracket of Equation 78. This
way, the behaviour of the code can be tuned.
10.5.2
Two examples
Figures 142 and 143 illustrate the function of the MATCH code for the case of 235 U(nth ,f)
and for the case of 241 Pu(n,f). The first case stands for a system with an almost completely
measured mass distribution. Only a few yields near symmetry need to be completed. The
MATCH code was used with its default options. The data for the second system are much
8
This weighting assumes that all values provided by the model have the same relevance for the complete
set of data. In detail this depends on the degree of correlations between the different variables, which
should be considered to develop a more adequate description for weighting. Moreover, one may also
arbitrarily increase the weight of the experimental data if desired, without loosing the correlations given
by the model for the unmeasured yields.
190
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General description of fission, GEF model, Validation and evaluation of nuclear data
Figure 142: Matching measured
235
U(nth ,f) mass yields with GEF results
Note: Adjustment of the fission-fragment mass yields from GEF (red symbols) to evaluated data
[141] (black symbols) with the MATCH code for the system 235 U(nth ,f). The blue symbols show
the set of fission yields that maximizes the combined likelihood function of the evaluated data
and the GEF result together with the covariance matrix from GEF. The upper figures show the
full mass distribution in linear and logarithmic scale. The lower figures zoom on specific parts
of the distribution. See text for details.
more incomplete. For example, there is a large gap around symmetry which is clearly seen
by the straight dashed line that connects the available experimental points. In this case,
the MATCH code was used with a relative weight of 100 for the experimental data and a
relative weight of 10 for the covariances.
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General description of fission, GEF model, 191
Validation and evaluation of nuclear data
Figure 143: Matching measured
241
Pu(nf ast ,f) mass yields with GEF results
Note: Adjustment of the fission-fragment mass yields from GEF (red symbols) to evaluated
data [141] (black symbols) with the MATCH code for the system 241 Pu(n,f), En = 2.5 MeV.
The blue symbols show the set of fission yields that maximises the combined likelihood function
of the evaluated data and the GEF result together with the covariance matrix from GEF. The
upper figures show the full mass distribution in linear and logarithmic scale. The lower figures
zoom on specific parts of the distribution. See text for details.
192
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General description of fission, GEF model, Conclusion
11
Conclusion
A new general approach for modelling nuclear fission using global theoretical models and
considerations on the basis of universal laws of physics and mathematics has been derived.
The most prominent features of the GEF model are the evolution of quantum-mechanical
wave functions in systems with complex shape, memory effects in the dynamics of stochastic processes, the influence of the Second Law of thermodynamics on the evolution of open
systems in terms of statistical mechanics, and the topological properties of a continuous
function in multi-dimensional space.
It has been demonstrated that the model reproduces the measured fission barriers and
the observed properties of the fission fragments, prompt neutrons and prompt-gamma radiation with a remarkable precision. This success reveals that the fission process possesses
a high degree of inherent regularity that is hardly deducible from microscopic models. The
suitability of the model for the evaluation of nuclear data is shown.
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General description of fission, GEF model, 193
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